A000984 -id:A000984 - cp's OEIS frontend

A005259 Apery (Apéry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2.

1, 5, 73, 1445, 33001, 819005, 21460825, 584307365, 16367912425, 468690849005, 13657436403073, 403676083788125, 12073365010564729, 364713572395983725, 11111571997143198073, 341034504521827105445, 10534522198396293262825, 327259338516161442321485

Offset: 0

Simon Plouffe, N. J. A. Sloane, May 20 1991

Conjecture: For each n = 1,2,3,... the Apéry polynomial A_n(x) = Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)^2*x^k is irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 21 2013
The expansions of exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 5*x + 49*x^2 + 685*x^3 + 11807*x^4 + 232771*x^5 + ... and exp( Sum_{n >= 1} a(n-1)*x^n/n ) = 1 + 3*x + 27*x^2 + 390*x^3 + 7038*x^4 + 144550*x^5 + ... both appear to have integer coefficients. See A267220. - Peter Bala, Jan 12 2016
Diagonal of the rational function R(x, y, z, w) = 1 / (1 - (w*x*y*z + w*x*y + w*z + x*y + x*z + y + z)); also diagonal of rational function H(x, y, z, w) = 1/(1 - w*(1+x)*(1+y)*(1+z)*(x*y*z + y*z + y + z + 1)). - Gheorghe Coserea, Jun 26 2018
Named after the French mathematician Roger Apéry (1916-1994). - Amiram Eldar, Jun 10 2021

			G.f. = 1 + 5*x + 73*x^2 + 1445*x^3 + 33001*x^4 + 819005*x^5 + 21460825*x^6 + ...
a(2) = (binomial(2,0) * binomial(2+0,0))^2 + (binomial(2,1) * binomial(2+1,1))^2 + (binomial(2,2) * binomial(2+2,2))^2 = (1*1)^2 + (2*3)^2 + (1*6)^2 = 1 + 36 + 36 = 73. - _Michael B. Porter_, Jul 14 2016

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 137-153.
Wolfram Koepf, Hypergeometric Identities. Ch. 2 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 55, 119 and 146, 1998.
Maxim Kontsevich and Don Zagier, Periods, pp. 771-808 of B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001.
Leonard Lipshitz and Alfred van der Poorten, "Rational functions, diagonals, automata and arithmetic." In Number Theory, Richard A. Mollin, ed., Walter de Gruyter, Berlin (1990), pp. 339-358.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..656 (first 101 terms from T. D. Noe)
Boris Adamczewski, Jason P. Bell and Eric Delaygue, Algebraic independence of G-functions and congruences "a la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
Jean-Paul Allouche, A remark on Apéry's numbers, J. Comput. Appl. Math., Vol. 83 (1997), pp. 123-125.
Roger Apéry, Irrationalité de zeta(2) et zeta(3), in Journées Arith. de Luminy. Colloque International du Centre National de la Recherche Scientifique (CNRS) held at the Centre Universitaire de Luminy, Luminy, Jun 20-24, 1978. Astérisque, Vol. 61 (1979), pp. 11-13.
Roger Apéry, Sur certaines séries entières arithmétiques, Groupe de travail d'analyse ultramétrique, Vol. 9, No. 1 (1981-1982), Exp. No. 16, 2 p.
Roger Apéry, Interpolation de fractions continues et irrationalité de certaines constantes, Bulletin de la section des sciences du C.T.H.S III (1981), pp. 37-53.
Thomas Baruchel and Carsten Elsner, On error sums formed by rational approximations with split denominators, arXiv preprint arXiv:1602.06445 [math.NT], 2016.
Frits Beukers, Another congruence for the Apéry numbers, J. Number Theory, Vol. 25, No. 2 (1987), pp. 201-210.
Frits Beukers, Consequences of Apéry's work on zeta(3), in "Zeta(3) irrationnel: les retombées", Rencontres Arithmétiques de Caen, June 2-3, 1995 [Mentions divisibility of a(n) by powers of 5 and powers of 11]
Francis Brown, Irrationality proofs for zeta values, moduli spaces and dinner parties, arXiv:1412.6508 [math.NT], 2014.
William Y. C. Chen, Qing-Hu Hou and Yan-Ping Mu, A telescoping method for double summations, J. Comp. Appl. Math., Vol. 196, No. 2 (2006), pp. 553-566, Example 4.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023. See Table 2 p. 7.
M. Coster, Email, Nov 1990
Eric Delaygue, Arithmetic properties of Apéry-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004.
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.
Gerald A. Edgar, A formula with Legendre polynomials, Sci. Math. Research posting Mar 21 2005.
G. A. Edgar, The Apéry Numbers as a Stieltjes Moment Sequence, arXiv:2005.10733 [math.CA], 2020.
Carsten Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., Vol. 43, No. 1 (2005), pp. 31-45.
Carsten Elsner, On prime-detecting sequences from Apéry's recurrence formulas for zeta(3) and zeta(2), JIS, Vol. 11 (2008), Article 08.5.1.
Stéphane Fischler, Irrationalité de valeurs de zeta, arXiv:math/0303066 [math.NT], 2003.
Scott Garrabrant and Igor Pak, Counting with irrational tiles, arXiv:1407.8222 [math.CO], 2014.
Ira Gessel, Some congruences for Apéry numbers, Journal of Number Theory, Vol. 14, No. 3 (1982) 362-368.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See gamma p. 3.
Richard K. Guy, Letter to N. J. A. Sloane, Oct 1985
Vaclav Kotesovec, Asymptotic of generalized Apéry sequences with powers of binomial coefficients, Nov 04 2012
Antoine Labelle et al., A formula for Apéry numbers, MathOverflow, 2025.
Leonard Lipshitz and Alfred J. van der Poorten, Rational functions, diagonals, automata and arithmetic, in: Richard A. Mollin (ed.), Number theory, Proceedings of the First Conference of the Canadian Number Theory Association Held at the Banff Center, Banff, Alberta, April 17-27, 1988, de Gruyter, 2016, pp. 339-358; alternative link; Wayback Machine copy.
Ji-Cai Liu, Supercongruences for the (p-1)th Apéry number, arXiv:1803.11442 [math.NT], 2018.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, Vol. 2 (2016), Article 5.
Stephen Melczer and Bruno Salvy, Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables, arXiv:1605.00402 [cs.SC], 2016.
Romeo Meštrović, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Robert Osburn and Brundaban Sahu, A supercongruence for generalized Domb numbers, Funct. Approx. Comment. Math., Vol. 48, No. 1 (2013), pp. 29-36; alternative link.
Math Overflow, A conjectured formula for Apéry numbers
Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, Journal de théorie des nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288; arXiv preprint, arXiv:1310.8635 [math.NT], 2013-2014.
Eric Rowland, Reem Yassawi and Christian Krattenthaler, Lucas congruences for the Apéry numbers modulo p^2, arXiv:2005.04801 [math.NT], 2020.
Andrew Strangeway, A Reconstruction Theorem for Quantum Cohomology of Fano Bundles on Projective Space, arXiv preprint arXiv:1302.5089 [math.AG], 2013.
Andrew Strangeway, Quantum reconstruction for Fano bundles on projective space, Nagoya Math. J., Vol. 218 (2015), pp. 1-28.
Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; arXiv preprint, arXiv:1401.0854 [math.NT], 2014.
Volker Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
Volker Strehl, Binomial identities -- combinatorial and algorithmic aspects, Discrete Mathematics, Vol. 136 (1994), 309-346.
Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.
Zhi-Wei Sun, Congruences for Franel numbers, arXiv preprint arXiv:1112.1034 [math.NT], 2011.
Zhi-Wei Sun, On sums of Apéry polynomials and related congruences, J. Number Theory 132(2012), 2673-2699. [_Zhi-Wei Sun_, Mar 21 2013]
Zhi-Wei Sun. Sun, On sums of Apéry polynomials and related congruences, arXiv:1101.1946 [math.NT], 2011-2014. [_Zhi-Wei Sun_, Mar 21 2013]
Alfred van der Poorten, A proof that Euler missed ..., Math. Intelligencer, Vol. 1, No. 4 (December 1979), pp. 196-203, (b_n) after eq. (1.2), and Exercise 3.
Chen Wang, Two congruences concerning Apéry numbers, arXiv:1909.08983 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Apéry Number.
Eric Weisstein's World of Mathematics, Strehl Identities.
Eric Weisstein's World of Mathematics, Schmidt's Problem.
Ernest X. W. Xia and Olivia X. M. Yao, A Criterion for the Log-Convexity of Combinatorial Sequences, The Electronic Journal of Combinatorics, Vol. 20 (2013), #P3.

Apéry's number or Apéry's constant zeta(3) is A002117. - N. J. A. Sloane, Jul 11 2023
Cf. A002736, A005258, A005429, A005430, A059415, A059416, A063007, A000172.
Cf. A006221, A000578, A006353.
Related to diagonal of rational functions: A268545-A268555.
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.
Cf. A092826 (prime terms).

List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*Binomial(n+k,k)^2)); # Muniru A Asiru, Sep 28 2018

a005259 n = a005259_list !! n
a005259_list = 1 : 5 : zipWith div (zipWith (-)
   (tail $ zipWith (*) a006221_list a005259_list)
   (zipWith (*) (tail a000578_list) a005259_list)) (drop 2 a000578_list)
-- Reinhard Zumkeller, Mar 13 2014

[&+[Binomial(n, k) ^2 *Binomial(n+k, k)^2: k in [0..n]]:n in  [0..17]]; // Marius A. Burtea, Jan 20 2020

a := proc(n) option remember; if n=0 then 1 elif n=1 then 5 else (n^(-3))* ( (34*(n-1)^3 + 51*(n-1)^2 + 27*(n-1) +5)*a((n-1)) - (n-1)^3*a((n-1)-1)); fi; end;
# Alternative:
a := n -> hypergeom([-n, -n, 1+n, 1+n], [1, 1, 1], 1):
seq(simplify(a(n)), n=0..17); # Peter Luschny, Jan 19 2020

Table[HypergeometricPFQ[{-n, -n, n+1, n+1}, {1,1,1}, 1],{n,0,13}] (* Jean-François Alcover, Apr 01 2011 *)
Table[Sum[(Binomial[n,k]Binomial[n+k,k])^2,{k,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 15 2011 *)
a[ n_] := SeriesCoefficient[ SeriesCoefficient[ SeriesCoefficient[ SeriesCoefficient[ 1 / (1 - t (1 + x ) (1 + y ) (1 + z ) (x y z + (y + 1) (z + 1))), {t, 0, n}], {x, 0, n}], {y, 0, n}], {z, 0, n}]; (* Michael Somos, May 14 2016 *)

a(n)=sum(k=0,n,(binomial(n,k)*binomial(n+k,k))^2) \\ Charles R Greathouse IV, Nov 20 2012

def A005259(n):
    m, g = 1, 0
    for k in range(n+1):
        g += m
        m *= ((n+k+1)*(n-k))**2
        m //=(k+1)**4
    return g # Chai Wah Wu, Oct 02 2022

D-finite with recurrence (n+1)^3*a(n+1) = (34*n^3 + 51*n^2 + 27*n + 5)*a(n) - n^3*a(n-1), n >= 1.
Representation as a special value of the hypergeometric function 4F3, in Maple notation: a(n)=hypergeom([n+1, n+1, -n, -n], [1, 1, 1], 1), n=0, 1, ... - Karol A. Penson Jul 24 2002
a(n) = Sum_{k >= 0} A063007(n, k)*A000172(k). A000172 = Franel numbers. - Philippe Deléham, Aug 14 2003
G.f.: (-1/2)*(3*x - 3 + (x^2-34*x+1)^(1/2))*(x+1)^(-2)*hypergeom([1/3,2/3],[1],(-1/2)*(x^2 - 7*x + 1)*(x+1)^(-3)*(x^2 - 34*x + 1)^(1/2)+(1/2)*(x^3 + 30*x^2 - 24*x + 1)*(x+1)^(-3))^2. - Mark van Hoeij, Oct 29 2011
Let g(x, y) = 4*cos(2*x) + 8*sin(y)*cos(x) + 5 and let P(n,z) denote the Legendre polynomial of degree n. Then G. A. Edgar posted a conjecture of Alexandru Lupas that a(n) equals the double integral 1/(4*Pi^2)*int {y = -Pi..Pi} int {x = -Pi..Pi} P(n,g(x,y)) dx dy. (Added Jan 07 2015: Answered affirmatively in Math Overflow question 178790) - Peter Bala, Mar 04 2012; edited by G. A. Edgar, Dec 10 2016
a(n) ~ (1+sqrt(2))^(4*n+2)/(2^(9/4)*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Nov 01 2012
a(n) = Sum_{k=0..n} C(n,k)^2 * C(n+k,k)^2. - Joerg Arndt, May 11 2013
0 = (-x^2+34*x^3-x^4)*y''' + (-3*x+153*x^2-6*x^3)*y'' + (-1+112*x-7*x^2)*y' + (5-x)*y, where y is g.f. - Gheorghe Coserea, Jul 14 2016
From Peter Bala, Jan 18 2020: (Start)
a(n) = Sum_{0 <= j, k <= n} (-1)^(n+j) * C(n,k)^2 * C(n+k,k)^2 * C(n,j) * C(n+k+j,k+j).
a(n) = Sum_{0 <= j, k <= n} C(n,k) * C(n+k,k) * C(k,j)^3 (see Koepf, p. 55).
a(n) = Sum_{0 <= j, k <= n} C(n,k)^2 * C(n,j)^2 * C(3*n-j-k,2*n) (see Koepf, p. 119).
Diagonal coefficients of the rational function 1/((1 - x - y)*(1 - z - t) - x*y*z*t) (Straub, 2014). (End)
a(n) = [x^n] 1/(1 - x)*( Legendre_P(n,(1 + x)/(1 - x)) )^m at m = 2. At m = 1 we get the Apéry numbers A005258. - Peter Bala, Dec 22 2020
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*A108625(n, k). - Peter Bala, Jul 18 2024
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n,k)^2 * C(n,j)^2 * C(k+j,k), see Labelle et al. link. - Max Alekseyev, Mar 12 2025

A005809 a(n) = binomial(3n,n).

Original entry on oeis.org

1, 3, 15, 84, 495, 3003, 18564, 116280, 735471, 4686825, 30045015, 193536720, 1251677700, 8122425444, 52860229080, 344867425584, 2254848913647, 14771069086725, 96926348578605, 636983969321700, 4191844505805495, 27619435402363035

Offset: 0

N. J. A. Sloane

Number of paths in Z X Z starting at (0,0) and ending at (3n,0) using steps in {(1,1),(1,-2)}.
Number of even trees with 2n edges and one distinguished vertex. Even trees are rooted plane trees where every vertex (including root) has even degree.
Hankel transform is 3^n*A051255(n), where A051255 is the Hankel transform of C(3n,n)/(2n+1). - Paul Barry, Jan 21 2007
a(n) is the number of stack polyominoes inscribed in an (n+1) X (n+1) box. Equivalently, a(n) is the number of unimodal compositions with n+1 parts in which the maximum value of the parts is n+1. For instance, for n = 2, we have the following compositions: (3,3,3), (2,3,3), (1,3,3), (3,3,1), (3,3,2), (2,2,3), (1,2,3), (2,3,1), (1,1,3), (1,3,1), (3,1,1), (2,3,2), (1,3,2), (3,2,1), (3,2,2). - Emanuele Munarini, Apr 07 2011
Conjecture: a(n)==3 (mod n^3) iff n is an odd prime. - Gary Detlefs, Mar 23 2013. The congruence a(p) = binomial(3*p,p) = 3 (mod p^3) for odd prime p is a known generalization of Wolstenholme's theorem. See Mestrovic, Section 6, equation 35. - Peter Bala, Dec 28 2014
In general, C(k*n,n) = C(k*n-1,n-1)*C((k*n)^2,2)/(3*n*C(k*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014

			G.f. = 1 + 3*x + 15*x^2 + 84*x^3 + 495*x^4 + 3003*x^5 + 18564*x^6 + ... - _Michael Somos_, Jan 30 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000[Terms 0 to 100 computed by T. D. Noe; terms 101 to 1000 by G. C. Greubel, Jan 14 2017]
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Barry, On the Central Coefficients of Riordan Matrices, Journal of Integer Sequences, 16 (2013), Article 13.5.1.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Naiomi Tuere Cameron, Random walks, trees and extensions of Riordan group techniques, Dissertation, Howard University, 2002.
Tom C. Copeland, Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers
Maciej Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv:1410.5747 [math.CO], 2014.
Milan Janjic, Two Enumerative Functions.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 436.
Yaping Liu, On the Recursiveness of Pascal Sequences, Global J. of Pure and Appl. Math. (2022) Vol. 18, No. 1, 71-80.
Mathematics Stack Exchange, Ordinary generating function for binomial(3n,n), Nov 2013.
Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Wojciech Młotkowski and Karol A. Penson, Probability distributions with binomial moments, Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol. 17, No. 2 (2014), 1450014; arXiv preprint, arXiv:1309.0595 [math.PR], 2013.
Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood, Jacobi Polynomials and Congruences Involving Some Higher-Order Catalan Numbers and Binomial Coefficients, Journal of Integer Sequences, Vol. 18 (2015), Article 15.11.7.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, Journal of Integer Sequences, Vol. 4 (2001), Article 01.2.1.

binomial(k*n,n): A000984 (k = 2), A005810 (k = 4), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).

Cf. A001764, A007318, A110608, A229705, A254157, A268196.

a005809 n = a007318 (3*n) n  -- Reinhard Zumkeller, May 06 2012

[ Binomial(3*n,n): n in [0..150] ]; // Vincenzo Librandi, Apr 21 2011

A005809:=n->binomial(3*n,n); seq(A005809(n), n=0..40); # Wesley Ivan Hurt, Mar 21 2014

R[ z_ ] := ((2-18*z + 27*z^2 + 3^(3/2)*z^(3/2)*(27*z-4)^(1/2))/2)^(1/3); f[ z_ ] := ( (R[ z ])^3 + (1-3*z)*(R[ z ])^2 + (1-6*z)*R[ z ] )/( (R[ z ])^4 + (1-6*z)*(R[ z ])^2 + (6*z-1)^2 )
Table[Binomial[3*n,n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011 *)

makelist(binomial(3*n,n),n,0,100); /* Emanuele Munarini, Apr 07 2011 */

B(x):=(2/sqrt(3*x))*sin((1/3)*asin(sqrt(27*x/4)))-1;
taylor(x*diff(B(x),x)/B(x),x,0,10); /* Vladimir Kruchinin, Oct 02 2015 */

a(n)=binomial(3*n,n) \\ Charles R Greathouse IV, Nov 20 2012

[binomial(3*n,n) for n in range(0, 22)] # Zerinvary Lajos, Dec 16 2009

The g.f. R[ z_ ] below (in the Mathematica field) was found by Kurt Persson (kurt(AT)math.chalmers.se) and communicated by Einar Steingrimsson (einar(AT)math.chalmers.se).
Using Stirling's formula in A000142, it is easy to get the asymptotic expression a(n) ~ (1/2) * (27/4)^n / sqrt(Pi*n / 3). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
a(n) = Sum_{k=0..n} C(n, k)*C(2n, k). - Paul Barry, May 15 2003
G.f.: 1/(1-3zg^2), where g=g(z) is given by g=1+zg^3, g(0)=1, i.e., (in Maple notation) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003
G.f.: x*B'(x)/B(x), where B(x)+1 is the g.f. for A001764. -  Vladimir Kruchinin, Oct 02 2015
a(n) ~ (1/2)*3^(1/2)*Pi^(-1/2)*n^(-1/2)*2^(-2*n)*3^(3*n)*(1 - 7/72*n^-1 + 49/10368*n^-2 + 6425/2239488*n^-3 - ...). - Joe Keane (jgk(AT)jgk.org), Nov 07 2003
a(n) = A006480(n)/A000984(n). - Lior Manor, May 04 2004
a(n) = Sum_{i_1=0..n, i_2=0..n} binomial(n, i_1)*binomial(n, i_2)*binomial(n, i_1+i_2). - Benoit Cloitre, Oct 14 2004
a(n) = Sum_{k=0..n} A109971(k)*3^k; a(0)=1, a(n) = Sum_{k=0..n} 3^k*C(3n-k,n-k)2k/(3n-k), n>0. - Paul Barry, Jan 21 2007
a(n) = A085478(2n,n). - Philippe Deléham, Sep 17 2009
E.g.f.: 2F2(1/3,2/3;1/2,1;27*x/4), where F(a1,a2;b1,b2;z) is a hypergeometric series. - Emanuele Munarini, Apr 12 2011
a(n) = Sum_{k=0..n} binomial(2*n+k-1,k). - Arkadiusz Wesolowski, Apr 02 2012
G.f.: cos((1/3)*asin(sqrt(27x/4)))/sqrt(1-27x/4). - Tom Copeland, May 24 2012
G.f.: A(x) = 1 + 6*x/(G(0)-6*x) where G(k) = (2*k+2)*(2*k+1) + 3*x*(3*k+1)*(3*k+2) - 6*x*(k+1)*(2*k+1)*(3*k+4)*(3*k+5)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jun 30 2012
D-finite with recurrence: 2*n*(2*n-1)*a(n) - 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013
a(n) = (2n+1)*A001764(n). - Johannes W. Meijer, Aug 22 2013
a(n) = C(3*n-1,n-1)*C(9*n^2,2)/(3*n*C(3*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014
a(n) = [x^n] 1/(1 - x)^(2*n+1). - Ilya Gutkovskiy, Oct 03 2017
a(n) = hypergeom([-2*n, -n], [1], 1). - Peter Luschny, Mar 19 2018
a(n) = Sum_{k=0..n} binomial(n, k) * binomial(2*n, n-k) = row sums of A110608. - Michael Somos, Jan 30 2019
0 = a(n)*(-3188646*a(n+2) +7322076*a(n+3) -2805111*a(n+4) +273585*a(n+5)) +a(n+1)*(+413343*a(n+2) -1252017*a(n+3) +538344*a(n+4) -55940*a(n+5)) +a(n+2)*(-4131*a(n+2) +38733*a(n+3) -21628*a(n+4) +2528*a(n+5)) for all n in Z. - Michael Somos, Jan 30 2019
Sum_{n>=1} 1/a(n) = A229705. - Amiram Eldar, Nov 14 2020
From Peter Bala, Feb 20 2022: (Start)
The o.g.f. A(x) satisfies the differential equation (4*x - 27*x^2)*A''(x) + (2 - 54*x)*A'(x) - 6*A(x) = 0, with A(0) = 1 and A'(0) = 3.
Algebraic equation: (1 - A(x))*(1 + 2*A(x))^2 +  27*x*A(x)^3 = 0.
Sum_{n >= 1} a(n)*( x*(2*x + 3)^2/(27*(1 + x)^3) )^n = x. (End)
From Vaclav Kotesovec, May 13 2022: (Start)
Sum_{n>=0} a(n) / 3^(2*n) = 2*cos(Pi/9).
Sum_{n>=0} a(n) / (27/2)^n = (1 + sqrt(3))/2.
Sum_{n>=0} a(n) / 3^(3*n) = 2*cos(Pi/18) / sqrt(3).
In general, for k > 27/4, Sum_{n>=0} a(n)/k^n = 2*cos(arccos(1 - 27/(2*k))/6) / sqrt(4 - 27/k). (End)
G.f.: hypergeom([1/3, 2/3], [1/2], 27*z/4), the Gauss hypergeometric function 2F1. - Karol A. Penson, Dec 12 2023
a(n) = 1/4^n * Sum_{k = n..3*n} binomial(k, n)*binomial(3*n, k). - Peter Bala, Jun 29 2025
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(4*n+1, k)*binomial(2*n-k, n-k). - Peter Bala, Sep 04 2025

A039599 Triangle formed from even-numbered columns of triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 9, 5, 1, 14, 28, 20, 7, 1, 42, 90, 75, 35, 9, 1, 132, 297, 275, 154, 54, 11, 1, 429, 1001, 1001, 637, 273, 77, 13, 1, 1430, 3432, 3640, 2548, 1260, 440, 104, 15, 1, 4862, 11934, 13260, 9996, 5508, 2244, 663, 135, 17, 1

Offset: 0

N. J. A. Sloane

T(n,k) is the number of lattice paths from (0,0) to (n,n) with steps E = (1,0) and N = (0,1) which touch but do not cross the line x - y = k and only situated above this line; example: T(3,2) = 5 because we have EENNNE, EENNEN, EENENN, ENEENN, NEEENN. - Philippe Deléham, May 23 2005
The matrix inverse of this triangle is the triangular matrix T(n,k) = (-1)^(n+k)* A085478(n,k). - Philippe Deléham, May 26 2005
Essentially the same as A050155 except with a leading diagonal A000108 (Catalan numbers) 1, 1, 2, 5, 14, 42, 132, 429, .... - Philippe Deléham, May 31 2005
Number of Grand Dyck paths of semilength n and having k downward returns to the x-axis. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)). Example: T(3,2)=5 because we have u(d)uud(d),uud(d)u(d),u(d)u(d)du,u(d)duu(d) and duu(d)u(d) (the downward returns to the x-axis are shown between parentheses). - Emeric Deutsch, May 06 2006
Riordan array (c(x),x*c(x)^2) where c(x) is the g.f. of A000108; inverse array is (1/(1+x),x/(1+x)^2). - Philippe Deléham, Feb 12 2007
The triangle may also be generated from M^n*[1,0,0,0,0,0,0,0,...], where M is the infinite tridiagonal matrix with all 1's in the super and subdiagonals and [1,2,2,2,2,2,2,...] in the main diagonal. - Philippe Deléham, Feb 26 2007
Inverse binomial matrix applied to A124733. Binomial matrix applied to A089942. - Philippe Deléham, Feb 26 2007
Number of standard tableaux of shape (n+k,n-k). - Philippe Deléham, Mar 22 2007
From Philippe Deléham, Mar 30 2007: (Start)
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1. Other triangles arise by choosing different values for (x,y):
(0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970
(1,0) -> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877;
(1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598;
(2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954;
(3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791;
(4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. (End)
The table U(n,k) = Sum_{j=0..n} T(n,j)*k^j is given in A098474. - Philippe Deléham, Mar 29 2007
Sequence read mod 2 gives A127872. - Philippe Deléham, Apr 12 2007
Number of 2n step walks from (0,0) to (2n,2k) and consisting of step u=(1,1) and d=(1,-1) and the path stays in the nonnegative quadrant. Example: T(3,0)=5 because we have uuuddd, uududd, ududud, uduudd, uuddud; T(3,1)=9 because we have uuuudd, uuuddu, uuudud, ududuu, uuduud, uduudu, uudduu, uduuud, uududu; T(3,2)=5 because we have uuuuud, uuuudu, uuuduu, uuduuu, uduuuu; T(3,3)=1 because we have uuuuuu. - Philippe Deléham, Apr 16 2007, Apr 17 2007, Apr 18 2007
Triangular matrix, read by rows, equal to the matrix inverse of triangle A129818. - Philippe Deléham, Jun 19 2007
Let Sum_{n>=0} a(n)*x^n = (1+x)/(1-mx+x^2) = o.g.f. of A_m, then Sum_{k=0..n} T(n,k)*a(k) = (m+2)^n. Related expansions of A_m are: A099493, A033999, A057078, A057077, A057079, A005408, A002878, A001834, A030221, A002315, A033890, A057080, A057081, A054320, A097783, A077416, A126866, A028230, A161591, for m=-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, respectively. - Philippe Deléham, Nov 16 2009
The Kn11, Kn12, Fi1 and Fi2 triangle sums link the triangle given above with three sequences; see the crossrefs. For the definitions of these triangle sums, see A180662. - Johannes W. Meijer, Apr 20 2011
4^n = (n-th row terms) dot (first n+1 odd integer terms). Example: 4^4 = 256 = (14, 28, 20, 7, 1) dot (1, 3, 5, 7, 9) = (14 + 84 + 100 + 49 + 9) = 256. - Gary W. Adamson, Jun 13 2011
The linear system of n equations with coefficients defined by the first n rows solve for diagonal lengths of regular polygons with N= 2n+1 edges; the constants c^0, c^1, c^2, ... are on the right hand side, where c = 2 + 2*cos(2*Pi/N). Example: take the first 4 rows relating to the 9-gon (nonagon), N = 2*4 + 1; with c = 2 + 2*cos(2*Pi/9) = 3.5320888.... The equations are (1,0,0,0) = 1; (1,1,0,0) = c; (2,3,1,0) = c^2; (5,9,5,1) = c^3. The solutions are 1, 2.53208..., 2.87938..., and 1.87938...; the four distinct diagonal lengths of the 9-gon (nonagon) with edge = 1. (Cf. comment in A089942 which uses the analogous operations but with c = 1 + 2*cos(2*Pi/9).) - Gary W. Adamson, Sep 21 2011
Also called the Lobb numbers, after Andrew Lobb, are a natural generalization of the Catalan numbers, given by L(m,n)=(2m+1)*Binomial(2n,m+n)/(m+n+1), where n >= m >= 0. For m=0, we get the n-th Catalan number. See added reference. - Jayanta Basu, Apr 30 2013
From Wolfdieter Lang, Sep 20 2013: (Start)
T(n, k) = A053121(2*n, 2*k). T(n, k) appears in the formula for the (2*n)-th power of the algebraic number rho(N):= 2*cos(Pi/N) = R(N, 2) in terms of the odd-indexed diagonal/side length ratios R(N, 2*k+1) = S(2*k, rho(N)) in the regular N-gon inscribed in the unit circle (length unit 1). S(n, x) are Chebyshev's S polynomials (see A049310):
  rho(N)^(2*n) = Sum_{k=0..n} T(n, k)*R(N, 2*k+1), n >= 0, identical in N > = 1. For a proof see the Sep 21 2013 comment under A053121. Note that this is the unreduced version if R(N, j) with j > delta(N), the degree of the algebraic number rho(N) (see A055034), appears.
  For the odd powers of rho(n) see A039598. (End)
Unsigned coefficients of polynomial numerators of Eqn. 2.1 of the Chakravarty and Kodama paper, defining the polynomials of A067311. - Tom Copeland, May 26 2016
The triangle is the Riordan square of the Catalan numbers in the sense of A321620. - Peter Luschny, Feb 14 2023

			Triangle T(n, k) begins:
  n\k     0     1     2     3     4     5    6   7   8  9
  0:      1
  1:      1     1
  2:      2     3     1
  3:      5     9     5     1
  4:     14    28    20     7     1
  5:     42    90    75    35     9     1
  6:    132   297   275   154    54    11    1
  7:    429  1001  1001   637   273    77   13   1
  8:   1430  3432  3640  2548  1260   440  104  15   1
  9:   4862 11934 13260  9996  5508  2244  663 135  17  1
  ... Reformatted by _Wolfdieter Lang_, Dec 21 2015
From _Paul Barry_, Feb 17 2011: (Start)
Production matrix begins
  1, 1,
  1, 2, 1,
  0, 1, 2, 1,
  0, 0, 1, 2, 1,
  0, 0, 0, 1, 2, 1,
  0, 0, 0, 0, 1, 2, 1,
  0, 0, 0, 0, 0, 1, 2, 1 (End)
From _Wolfdieter Lang_, Sep 20 2013: (Start)
Example for rho(N) = 2*cos(Pi/N) powers:
n=2: rho(N)^4 = 2*R(N,1) + 3*R(N,3) + 1*R(N, 5) =
  2 + 3*S(2, rho(N)) + 1*S(4, rho(N)), identical in N >= 1. For N=4 (the square with only one distinct diagonal), the degree delta(4) = 2, hence R(4, 3) and R(4, 5) can be reduced, namely to R(4, 1) = 1 and R(4, 5) = -R(4,1) = -1, respectively. Therefore, rho(4)^4 =(2*cos(Pi/4))^4 = 2 + 3 -1 = 4. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
T. Myers and L. Shapiro, Some applications of the sequence 1, 5, 22, 93, 386, ... to Dyck paths and ordered trees, Congressus Numerant., 204 (2010), 93-104.

T. D. Noe, Rows n=0..50 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Quang T. Bach and Jeffrey B. Remmel, Generating functions for descents over permutations which avoid sets of consecutive patterns, arXiv:1510.04319 [math.CO], 2015 (see p.25).
M. Barnabei, F. Bonetti and M. Silimbani, Two permutation classes enumerated by the central binomial coefficients, arXiv preprint arXiv:1301.1790 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.3.8
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010), Article 10.8.2, example 15.
Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
Paul Barry, Comparing two matrices of generalized moments defined by continued fraction expansions, arXiv preprint arXiv:1311.7161 [math.CO], 2013 and J. Int. Seq. 17 (2014) # 14.5.1.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 15, 29.
Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 12.
Jonathan E. Beagley and Paul Drube, Combinatorics of Tableau Inversions, Electron. J. Combin., 22 (2015), #P2.44.
S. Chakravarty and Y. Kodama, A generating function for the N-soliton solutions of the Kadomtsev-Petviashvili II equation, arXiv preprint arXiv:0802.0524v2 [nlin.SI], 2008.
Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, J. Int. Seqs., Vol. 21 (2018), #18.2.1.
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 11.
Paul Drube, Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers, arXiv:1606.04869 [math.CO], 2016.
Paul Drube, Generalized Path Pairs and Fuss-Catalan Triangles, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8.
T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, example p 32.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Thomas Koshy, Lobb's generalization of Catalan's parenthesization problem, The College Mathematics Journal 40 (2), March 2009, 99-107, DOI:10.1080/07468342.2009.11922344.
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 11.
Andrew Lobb, Deriving the n-th Catalan number, Mathematical Gazette, Vol. 83, No. 496 (March 1999), 109-110.
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174. See page 161.
Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016 (see 2.8).
A. Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14, No. 4 (1957), 405-414: 124. [Note: there is a typo]
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp. 29 (129) (1975) 215-222
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33
Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
Yidong Sun and Luping Ma, Minors of a class of Riordan arrays related to weighted partial Motzkin paths. Eur. J. Comb. 39, 157-169 (2014), Table 2.2.
Wikipedia, Lobb number
W.-J. Woan, L. Shapiro and D. G. Rogers, The Catalan numbers, the Lebesgue integral and 4^{n-2}, Amer. Math. Monthly, 104 (1997), 926-931.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017), 3081-3091.

Columns give: A000108, A000245, A000344, A000588, A001392, A000589, A000590, A000012, A005408, A014107(n>1).
Row sums: A000984.
Triangle sums (see the comments): A000958 (Kn11), A001558 (Kn12), A088218 (Fi1, Fi2).
Cf. A008313, A039598, A084938, A000007, A067311, A321620.

/* As triangle */ [[Binomial(2*n, k+n)*(2*k+1)/(k+n+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 16 2015

T:=(n,k)->(2*k+1)*binomial(2*n,n-k)/(n+k+1): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form # Emeric Deutsch, May 06 2006
T := proc(n, k) option remember; if k = n then 1 elif k > n then 0 elif k = 0 then T(n-1, 0) + T(n-1,1) else T(n-1, k-1) + 2*T(n-1, k) + T(n-1, k+1) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..9) od; # Peter Luschny, Feb 14 2023

Table[Abs[Differences[Table[Binomial[2 n, n + i], {i, 0, n + 1}]]], {n, 0,7}] // Flatten (* Geoffrey Critzer, Dec 18 2011 *)
Join[{1},Flatten[Table[Binomial[2n-1,n-k]-Binomial[2n-1,n-k-2],{n,10},{k,0,n}]]] (* Harvey P. Dale, Dec 18 2011 *)
Flatten[Table[Binomial[2*n,m+n]*(2*m+1)/(m+n+1),{n,0,9},{m,0,n}]] (* Jayanta Basu, Apr 30 2013 *)

a(n, k) = (2*n+1)/(n+k+1)*binomial(2*k, n+k)
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(a(y, x), ", ")); print(""))
trianglerows(10) \\ Felix Fröhlich, Jun 24 2016

# Algorithm of L. Seidel (1877)
# Prints the first n rows of the triangle
def A039599_triangle(n) :
    D = [0]*(n+2); D[1] = 1
    b = True ; h = 1
    for i in range(2*n-1) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        if b : print([D[z] for z in (1..h-1)])
        b = not b
A039599_triangle(10)  # Peter Luschny, May 01 2012

T(n,k) = C(2*n-1, n-k) - C(2*n-1, n-k-2), n >= 1, T(0,0) = 1.
From Emeric Deutsch, May 06 2006: (Start)
T(n,k) = (2*k+1)*binomial(2*n,n-k)/(n+k+1).
G.f.: G(t,z)=1/(1-(1+t)*z*C), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function. (End)
The following formulas were added by Philippe Deléham during 2003 to 2009: (Start)
Triangle T(n, k) read by rows; given by A000012 DELTA A000007, where DELTA is Deléham's operator defined in A084938.
T(n, k) = C(2*n, n-k)*(2*k+1)/(n+k+1). Sum(k>=0; T(n, k)*T(m, k) = A000108(n+m)); A000108: numbers of Catalan.
T(n, 0) = A000108(n); T(n, k) = 0 if k>n; for k>0, T(n, k) = Sum_{j=1..n} T(n-j, k-1)*A000108(j).
T(n, k) = A009766(n+k, n-k) = A033184(n+k+1, 2k+1).
G.f. for column k: Sum_{n>=0} T(n, k)*x^n = x^k*C(x)^(2*k+1) where C(x) = Sum_{n>=0} A000108(n)*x^n is g.f. for Catalan numbers, A000108.
T(0, 0) = 1, T(n, k) = 0 if n<0 or n=1, T(n, k) = T(n-1, k-1) + 2*T(n-1, k) + T(n-1, k+1).
a(n) + a(n+1) = 1 + A000108(m+1) if n = m*(m+3)/2; a(n) + a(n+1) = A039598(n) otherwise.
T(n, k) = A050165(n, n-k).
Sum_{j>=0} T(n-k, j)*A039598(k, j) = A028364(n, k).
Matrix inverse of the triangle T(n, k) = (-1)^(n+k)*binomial(n+k, 2*k) = (-1)^(n+k)*A085478(n, k).
Sum_{k=0..n} T(n, k)*x^k = A000108(n), A000984(n), A007854(n), A076035(n), A076036(n) for x = 0, 1, 2, 3, 4.
Sum_{k=0..n} (2*k+1)*T(n, k) = 4^n.
T(n, k)*(-2)^(n-k) = A114193(n, k).
Sum_{k>=h} T(n,k) = binomial(2n,n-h).
Sum_{k=0..n} T(n,k)*5^k = A127628(n).
Sum_{k=0..n} T(n,k)*7^k = A115970(n).
T(n,k) = Sum_{j=0..n-k} A106566(n+k,2*k+j).
Sum_{k=0..n} T(n,k)*6^k = A126694(n).
Sum_{k=0..n} T(n,k)*A000108(k) = A007852(n+1).
Sum_{k=0..floor(n/2)} T(n-k,k) = A000958(n+1).
Sum_{k=0..n} T(n,k)*(-1)^k = A000007(n).
Sum_{k=0..n} T(n,k)*(-2)^k = (-1)^n*A064310(n).
T(2*n,n) = A126596(n).
Sum_{k=0..n} T(n,k)*(-x)^k = A000007(n), A126983(n), A126984(n), A126982(n), A126986(n), A126987(n), A127017(n), A127016(n), A126985(n), A127053(n) for x=1,2,3,4,5,6,7,8,9,10 respectively.
Sum_{j>=0} T(n,j)*binomial(j,k) = A116395(n,k).
T(n,k) = Sum_{j>=0} A106566(n,j)*binomial(j,k).
T(n,k) = Sum_{j>=0} A127543(n,j)*A038207(j,k).
Sum_{k=0..floor(n/2)} T(n-k,k)*A000108(k) = A101490(n+1).
T(n,k) = A053121(2*n,2*k).
Sum_{k=0..n} T(n,k)*sin((2*k+1)*x) = sin(x)*(2*cos(x))^(2*n).
T(n,n-k) = Sum_{j>=0} (-1)^(n-j)*A094385(n,j)*binomial(j,k).
Sum_{j>=0} A110506(n,j)*binomial(j,k) = Sum_{j>=0} A110510(n,j)*A038207(j,k) = T(n,k)*2^(n-k).
Sum_{j>=0} A110518(n,j)*A027465(j,k) = Sum_{j>=0} A110519(n,j)*A038207(j,k) = T(n,k)*3^(n-k).
Sum_{k=0..n} T(n,k)*A001045(k) = A049027(n), for n>=1.
Sum_{k=0..n} T(n,k)*a(k) = (m+2)^n if Sum_{k>=0} a(k)*x^k = (1+x)/(x^2-m*x+1).
Sum_{k=0..n} T(n,k)*A040000(k) = A001700(n).
Sum_{k=0..n} T(n,k)*A122553(k) = A051924(n+1).
Sum_{k=0..n} T(n,k)*A123932(k) = A051944(n).
Sum_{k=0..n} T(n,k)*k^2 = A000531(n), for n>=1.
Sum_{k=0..n} T(n,k)*A000217(k) = A002457(n-1), for n>=1.
Sum{j>=0} binomial(n,j)*T(j,k)= A124733(n,k).
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000984(n), A089022(n), A035610(n), A130976(n), A130977(n), A130978(n), A130979(n), A130980(n), A131521(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
Sum_{k=0..n} T(n,k)*A005043(k) = A127632(n).
Sum_{k=0..n} T(n,k)*A132262(k) = A089022(n).
T(n,k) + T(n,k+1) = A039598(n,k).
T(n,k) = A128899(n,k)+A128899(n,k+1).
Sum_{k=0..n} T(n,k)*A015518(k) = A076025(n), for n>=1. Also Sum_{k=0..n} T(n,k)*A015521(k) = A076026(n), for n>=1.
Sum_{k=0..n} T(n,k)*(-1)^k*x^(n-k) = A033999(n), A000007(n), A064062(n), A110520(n), A132863(n), A132864(n), A132865(n), A132866(n), A132867(n), A132869(n), A132897(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Sum_{k=0..n} T(n,k)*(-1)^(k+1)*A000045(k) = A109262(n), A000045:= Fibonacci numbers.
Sum_{k=0..n} T(n,k)*A000035(k)*A016116(k) = A143464(n).
Sum_{k=0..n} T(n,k)*A016116(k) = A101850(n).
Sum_{k=0..n} T(n,k)*A010684(k) = A100320(n).
Sum_{k=0..n} T(n,k)*A000034(k) = A029651(n).
Sum_{k=0..n} T(n,k)*A010686(k) = A144706(n).
Sum_{k=0..n} T(n,k)*A006130(k-1) = A143646(n), with A006130(-1)=0.
T(n,2*k)+T(n,2*k+1) = A118919(n,k).
Sum_{k=0..j} T(n,k) = A050157(n,j).
Sum_{k=0..2} T(n,k) = A026012(n); Sum_{k=0..3} T(n,k)=A026029(n).
Sum_{k=0..n} T(n,k)*A000045(k+2) = A026671(n).
Sum_{k=0..n} T(n,k)*A000045(k+1) = A026726(n).
Sum_{k=0..n} T(n,k)*A057078(k) = A000012(n).
Sum_{k=0..n} T(n,k)*A108411(k) = A155084(n).
Sum_{k=0..n} T(n,k)*A057077(k) = 2^n = A000079(n).
Sum_{k=0..n} T(n,k)*A057079(k) = 3^n = A000244(n).
Sum_{k=0..n} T(n,k)*(-1)^k*A011782(k) = A000957(n+1).
(End)
T(n,k) = Sum_{j=0..k} binomial(k+j,2j)*(-1)^(k-j)*A000108(n+j). - Paul Barry, Feb 17 2011
Sum_{k=0..n} T(n,k)*A071679(k+1) = A026674(n+1). - Philippe Deléham, Feb 01 2014
Sum_{k=0..n} T(n,k)*(2*k+1)^2 = (4*n+1)*binomial(2*n,n). - Werner Schulte, Jul 22 2015
Sum_{k=0..n} T(n,k)*(2*k+1)^3 = (6*n+1)*4^n. - Werner Schulte, Jul 22 2015
Sum_{k=0..n} (-1)^k*T(n,k)*(2*k+1)^(2*m) = 0 for 0 <= m < n (see also A160562). - Werner Schulte, Dec 03 2015
T(n,k) = GegenbauerC(n-k,-n+1,-1) - GegenbauerC(n-k-1,-n+1,-1). - Peter Luschny, May 13 2016
T(n,n-2) = A014107(n). - R. J. Mathar, Jan 30 2019
T(n,n-3) = n*(2*n-1)*(2*n-5)/3. - R. J. Mathar, Jan 30 2019
T(n,n-4) = n*(n-1)*(2*n-1)*(2*n-7)/6. - R. J. Mathar, Jan 30 2019
T(n,n-5) = n*(n-1)*(2*n-1)*(2*n-3)*(2*n-9)/30. - R. J. Mathar, Jan 30 2019

Corrected by Philippe Deléham, Nov 26 2009, Dec 14 2009

A002894 a(n) = binomial(2n, n)^2.

Original entry on oeis.org

1, 4, 36, 400, 4900, 63504, 853776, 11778624, 165636900, 2363904400, 34134779536, 497634306624, 7312459672336, 108172480360000, 1609341595560000, 24061445010950400, 361297635242552100, 5445717990022688400, 82358080713306090000, 1249287673091590440000

Offset: 0

N. J. A. Sloane

a(n) is the number of monotonic paths (only moving N and E) in the lattice [0..2n] X [0..2n] that contain the points (0,0), (n,n) and (2n,2n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004
Expansion of K(k) / (Pi/2) in powers of m/16 = (k/4)^2, where K(k) is the complete elliptic integral of the first kind evaluated at k. - Michael Somos, Mar 04 2003
Square lattice walks that start and end at origin after 2n steps. - Gareth McCaughan and Michael Somos, Jun 12 2004
If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic) then a(n)=E[(tr(A^k))^{2n}] for any k > 4. - Andrew V. Sutherland, Apr 01 2008
From R. H. Hardin, Feb 03 2016 and R. J. Mathar, Feb 18 2016: (Start)
Also, number of 2 X (2n) arrays of permutations of 2n copies of 0 or 1 with row sums equal.
For example, some solutions for n=3:
  0 1 0 1 0 1   0 1 0 1 1 0   0 0 1 0 1 1   1 1 1 0 0 0
  1 0 0 0 1 1   1 1 0 1 0 0   0 0 0 1 1 1   0 0 1 1 0 1
There is a simple combinatorial argument to show that this is a(n): We have 2n copies of 0's and 1's and need equal row sums. Therefore there must be n 1's in each of the two rows. Otherwise there are no constraints, so there are C(2n,n) ways of placing the 1's in the first row and independently C(2n,n) ways of placing the 1's in the second. The product is clearly C(2n,n)^2. (End)
Also the even part of the bisection of A241530. One half of the odd part is given in A000894. - Wolfdieter Lang, Sep 06 2016
From Peter Bala, Jan 26 2018: (Start)
Let S = {[1,0,0], [0,1,0], [1,0,1], [0,1,1]} be a set of four column vectors. Then a(n) equals the number of 3 X k arrays whose columns belong to the set S and whose row sums are all equal to n (apply Eger, Theorem 3). An example is given below. Equivalently, a(n) equals the number of lattice paths from (0,0,0) to (n,n,n) using steps (1,0,0), (0,1,0), (1,0,1) and (0,1,1).
The o.g.f. for the sequence equals the diagonal of the rational function 1/(1 - (x + y + x*z + y*z)).
Row sums of A069466. (End)
Also, the constant term in the expansion of (x + 1/x + y + 1/y)^(2n). - Christopher J. Smyth, Sep 26 2018
Number of ways to place 2n^2 nonattacking pawns on a 2n x 2n board. - Tricia Muldoon Brown, Dec 12 2018
For n>0, a(n) is the number of Littlewood polynomials of degree 4n-1 that have a closed Lill path. A polynomial p(x) has a closed Lill path if and only if p(x) is divisible by x^(2)+1. - Raul Prisacariu, Aug 28 2024

			G.f. = 1 + 4*x + 36*x^2 + 400*x^3 + 4900*x^4 + 63504*x^5 + 853776*x^6 + ... - _Michael Somos_, Aug 06 2014
From _Peter Bala_, Jan 26 2018: (Start)
a(2) = 36: The thirty six 3 x k arrays with columns belonging to the set of column vectors S = {[1,0,0], [0,1,0], [1,0,1], [0,1,1]} and having all row sums equal to 2 are the 6 distinct arrays obtained by permuting the columns of
  /1 1 0 0\
  |0 0 1 1|,
  \0 0 1 1/
the 6 distinct arrays obtained by permuting the columns of
  /0 0 1 1\
  |1 1 0 0|
  \0 0 1 1/
and the 24 arrays obtained by permuting the columns of
  /1 0 1 0\
  |0 1 0 1|. (End)
  \0 0 1 1/

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 591,828.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
Leonard Lipshitz and A. van der Poorten. "Rational functions, diagonals, automata and arithmetic." In Number Theory, Richard A. Mollin, ed., Walter de Gruyter, Berlin (1990): 339-358.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. Bacher, Meander algebras, Institut Fourier, 1999.
E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78.
Arnaud Beauville, Les familles stables de courbes elliptiques sur P^1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982, 657-660. MR0664643 (83h:14008)
Alin Bostan, Armin Straub, and Sergey Yurkevich, On the representability of sequences as constant terms, arXiv:2212.10116 [math.NT], 2022.
Tricia Muldoon Brown, The Problem of Pawns, The Electronic Journal of Combinatorics (2019) Vol. 26, Issue 3, #P3.21. Also arXiv:1811.09606, [math.CO], 2018.
John Maxwell Campbell, New series involving harmonic numbers and squared central binomial coefficients, Rocky Mountain J. Math., 49 (2019), 2513-2544.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Murray Elder, Cogrowth, 2011.
M. Elder, A. Rechnitzer, E. J. Janse van Rensburg, and T. Wong, The cogrowth series for BS(N,N) is D-finite, arXiv:1309.4184 [math.GR], 2013.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 90.
Florian Fürnsinn and Sergey Yurkevich, Algebraicity of hypergeometric functions with arbitrary parameters, arXiv:2308.12855 [math.CA], 2023.
Davidson Noby Joseph and Igor Boettcher, Walking on Archimedean Lattices: Insights from Bloch Band Theory, arXiv:2507.12662 [cond-mat.stat-mech], 2025. See p. 18.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2010.
Markus Kuba and Alois Panholzer, Lattice paths and the diagonal of the cube, arXiv:2411.03930 [math.CO], 2024. See p. 14.
L. Lipshitz and A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic
Raul Prisacariu, Littlewood Polynomials of Degree n with Closed Lill Paths
Eric M. Rains, High powers of random elements of compact Lie groups, Probability Theory and Related Fields 107 (1997), 219-241.
Grzegorz Siudem and Agata Fronczak, Bell polynomials in the series expansions of the Ising model, arXiv:2007.16132 [math-ph], 2020.
Eric Weisstein's World of Mathematics, Lattice Path.
D. Zagier, Integral solutions of Apéry-like recurrence equations. See line G in sporadic solutions table of page 5.

Row sums of A069466.
Row 2 of A268367 (even terms).
Equals 4*A060150.
Cf. A000984, A000515, A010370, A054474 (INVERTi transform), A172390, A000897, A002897, A006480, A008977, A186420, A188662, A000894, A241530, A002898 (walks hex lattice).

[Binomial(2*n, n)^2: n in [0..20]]; // Vincenzo Librandi, Aug 07 2014

Maple
```
A002894 := n-> binomial(2*n,n)^2.
```

CoefficientList[Series[Hypergeometric2F1[1/2, 1/2, 1, 16x], {x, 0, 20}], x]
Table[Binomial[2n,n]^2,{n,0,20}] (* Harvey P. Dale, Jul 06 2011 *)
a[ n_] := SeriesCoefficient[ EllipticK[16 x] / (Pi/2), {x, 0, n}]; (* Michael Somos, Aug 06 2014 *)
a[n_] := 16^n HypergeometricPFQ[{1/2, -2 n, 2 n + 1}, {1, 1}, 1];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 14 2018 *)

PARI
```
{a(n) = binomial(2*n, n)^2};
```

{a(n) = if( n<0, 0, polcoeff( polcoeff( polcoeff( 1 / (1 - x * (y + z + 1/y + 1/z)) + x * O(x^(2*n)), 2*n), 0), 0))}; /* Michael Somos, Jun 12 2004 */

[binomial(2*n, n)**2 for n in range(17)]  # Zerinvary Lajos, Apr 21 2009

D-finite with recurrence: (n+1)^2*a(n+1) = 4*(2*n + 1)^2*a(n). - Matthijs Coster, Apr 28 2004
a(n) ~ Pi^(-1)*n^(-1)*2^(4*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
G.f.: F(1/2, 1/2; 1; 16*x) = 1 / AGM(1, (1 - 16*x)^(1/2)) = K(4*sqrt(x)) / (Pi/2), where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre. - Michael Somos, Mar 04 2003
G.f.: 2*EllipticK(4*sqrt(x))/Pi, using Maple's convention for elliptic integrals.
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = BesselI(0, 2x)^2.
a(n) = A000984(n)^2. - Jonathan Vos Post, Jun 17 2007
E.g.f.: (BesselI(0, 2*x))^2 = 1+2*x^2/(U(0)-2*x^2); U(k) = 2*x^2*(2*k+1)+(k+1)^3-2*x^2*(2*k+3)*(k+1)^3/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2011
In generally, for (BesselI(b, 2x))^2=((x^(2*b))/(GAMMA(b+1))^2)*(1+(2*x^2)*(2*b+1)/(Q(0)-(2*x^2)*(2*b+1)); Q(k)=(2*x^2)*(2*k+2*b+1)+(k+1)*(k+b+1)*(k+2*b+1)-(2*x^2)*(k+1)*(k+b+1)*(k+2*b+1)*(2*k+2*b+3)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2011
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 4*(2*k+1)^2*x*(1+4*x)^2/(4*(2*k+1)^2*x*(1+4*x)^2 + (k+1)^2*(1+4*x)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 01 2013
0 = +a(n)*(+393216*a(n+2) -119040*a(n+3) +6860*a(n+4)) +a(n+1)*(-16128*a(n+2) +6928*a(n+3) -465*a(n+4)) +a(n+2)*(+36*a(n+2) -63*a(n+3) +6*a(n+4)) for all n in Z. - Michael Somos, Aug 06 2014
Integral representation as the n-th moment of a positive function W(x) on (0,16), in Maple notation, W(x) = EllipticK(sqrt(1-x/16))/(2*Pi^2*sqrt(x)); a(n) = Integral_{x=0..16} x^n*W(x) dx, n>=0. The function W(x) is singular at x=0 and W(16) = 1/(16*Pi). This representation is unique since W(x) is the solution of the Hausdorff moment problem. - Stanley Smith and Karol A. Penson, Jun 19 2015
a(n) ~ 16^n*(2-2/(8*n+2)^2+21/(8*n+2)^4-671/(8*n+2)^6+45081/(8*n+2)^8)^2/((4*n+1)* Pi). - Peter Luschny, Oct 14 2015
a(n) = binomial(2*n,n)*binomial(2*n,n) = ( [x^n](1 + x)^(2*n) ) *( [x^n](1 + x)^(2*n) ) = [x^n](F(x)^(4*n)), where F(x) = 1 + x + x^2 + 4*x^3 + 20*x^4 + 120*x^5 + 798*x^6 + 5697*x^7 + ... appears to have integer coefficients. For similar results see A000897, A002897, A006480, A008977, A186420 and A188662. - Peter Bala, Jul 14 2016
a(n) = Sum_{k = 0..n} binomial(2*n + k,k)*binomial(n,k)^2. Cf. A005258(n) = Sum_{k = 0..n} binomial(n + k,k)*binomial(n,k)^2. - Peter Bala, Jul 27 2016
a(n) = A241530(2*n), n >= 0. - Wolfdieter Lang, Sep 06 2016
E.g.f.: 2F2(1/2,1/2; 1,1; 16*x). - Ilya Gutkovskiy, Jan 23 2018
a(n) = 16^n*hypergeom([1/2, -2*n, 2*n + 1], [1, 1], 1). - Peter Luschny, Mar 14 2018
The right-hand side of the binomial coefficient identity Sum_{k = 0..n} C(n,k)*C(n+k,k)*C(2*n+2*k,n+k)*(-4)^(n-k) = a(n). - Peter Bala, Mar 16 2018
a(n) = [x^n] (1 - x)^(2*n) * P(2*n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Compare with A245086(n) = [x^n] (1 - x)^(2*n) * P(n,(1 + x)/(1 - x)). - Peter Bala, Mar 23 2022
a(n) = Sum_{k=0..n} multinomial(2n [k k (n-k) (n-k)]), which is another way to count random walks on Z^2, with steps of (0,+-1) or (+-1,0), that return to the point of origin after 2n steps (not necessarily for the first time), as is C(2n,n)^2. - Shel Kaphan, Jan 12 2023
0 = a(n)*(+393216*a(n+2) -119040*a(n+3) +6860*a(n+4)) +a(n+1)*(-16128*a(n+2) +6928*a(n+3) -465*a(n+4)) +a(n+2)*(+36*a(n+2) -63*a(n+3) +6*a(n+4)) for n>=0. - Michael Somos, May 30 2023
From Peter Bala, Sep 12 2023: (Start)
Right-hand side of the binomial coefficient identities
1) Sum_{k = 0..n} (-1)^(n+k) * C(n,k)*C(n+k,n)*C(2*n+k,n) = a(n).
2) 2*Sum_{k = 0..n} (-1)^(n+k) * C(n,k)*C(n+k-1,n)*C(2*n+k-1,n) = a(n) for n >= 1.
3) (4/3)*Sum_{k = 0..n} (-1)^(n+k) * C(n,k)*C(n+k,n)*C(2*n+k-1,n) = a(n) for n >= 1. (End)

Edited by N. J. A. Sloane, Feb 18 2016

A106529 Numbers having k prime factors (counted with multiplicity), the largest of which is the k-th prime.

Original entry on oeis.org

2, 6, 9, 20, 30, 45, 50, 56, 75, 84, 125, 126, 140, 176, 189, 196, 210, 264, 294, 315, 350, 396, 416, 440, 441, 490, 525, 594, 616, 624, 660, 686, 735, 875, 891, 924, 936, 968, 990, 1029, 1040, 1088, 1100, 1225, 1386, 1404, 1452, 1456, 1485, 1540, 1560

Offset: 1

Matthew Ryan (mattryan1994(AT)hotmail.com), May 30 2005

nonn

It seems that the ratio between successive terms tends to 1 as n increases, meaning perhaps that most numbers are in this sequence.
The number of terms that have the k-th prime as their largest prime factor is A000984(k), the k-th central binomial coefficient. E.g., 6 and 9 are the A000984(2)=2 terms in {a(n)} that have prime(2)=3 as their largest prime factor.
The sequence contains the positive integers m such that the rank of the partition B(m) = 0. For m >= 2, B(m) is defined as the partition obtained by taking the prime decomposition of m and replacing each prime factor p with its index i (i.e., i-th prime = p); also B(1) = the empty partition. For example, B(350) = B(2*5^2*7) = [1,3,3,4]. B is a bijection between the positive integers and the set of all partitions. The rank of a partition P is the largest part of P minus the number of parts of P. - Emeric Deutsch, May 09 2015
Also Heinz numbers of balanced partitions, counted by A047993. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Feb 08 2021

			a(7)=50 because 50=2*5*5 is, for k=3, the product of k primes, the largest of which is the k-th prime, and 50 is the 7th such number.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000984.
A001222 counts prime factors.
A056239 adds up prime indices.
A061395 selects maximum prime index.
A112798 lists the prime indices of each positive integer.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A090858 counts partitions of rank 1.
- A098124 counts balanced compositions.
- A340596 counts co-balanced factorizations.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
Cf. A006141, A064174, A096401, A117409, A168659, A324522, A340654, A340655, A340656, A340657.

with(numtheory): a := proc (n) options operator, arrow: pi(max(factorset(n)))-bigomega(n) end proc: A := {}: for i from 2 to 1600 do if a(i) = 0 then A := `union`(A, {i}) else  end if end do: A; # Emeric Deutsch, May 09 2015

Select[Range@ 1560, PrimePi@ FactorInteger[#][[-1, 1]] == PrimeOmega@ # &] (* Michael De Vlieger, May 09 2015 *)

For all terms, A001222(a(n)) = A061395(a(n)). - Gus Wiseman, Feb 08 2021

A009766 Catalan's triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e., T(n,k) = Sum_{j=0..k} T(n-1,j).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 9, 14, 14, 1, 5, 14, 28, 42, 42, 1, 6, 20, 48, 90, 132, 132, 1, 7, 27, 75, 165, 297, 429, 429, 1, 8, 35, 110, 275, 572, 1001, 1430, 1430, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934

Offset: 0

Wouter Meeussen

The entries in this triangle (in its many forms) are often called ballot numbers.
T(n,k) = number of standard tableaux of shape (n,k) (n > 0, 0 <= k <= n). Example: T(3,1) = 3 because we have 134/2, 124/3 and 123/4. - Emeric Deutsch, May 18 2004
T(n,k) is the number of full binary trees with n+1 internal nodes and jump-length k. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given ordered tree is called the jump-length. - Emeric Deutsch, Jan 18 2007
The k-th diagonal from the right (k=1, 2, ...) gives the sequence obtained by asking in how many ways can we toss a fair coin until we first get k more heads than tails. The k-th diagonal has formula k(2m+k-1)!/(m!(m+k)!) and g.f. (C(x))^k where C(x) is the generating function for the Catalan numbers, (1-sqrt(1-4*x))/(2*x). - Anthony C Robin, Jul 12 2007
T(n,k) is also the number of order-decreasing and order-preserving full transformations (of an n-element chain) of waist k (waist (alpha) = max(Im(alpha))). - Abdullahi Umar, Aug 25 2008
Formatted as an upper right triangle (see tables) a(c,r) is the number of different triangulated planar polygons with c+2 vertices, with triangle degree c-r+1 for the same vertex X (c=column number, r=row number, with c >= r >= 1). - Patrick Labarque, Jul 28 2010
The triangle sums, see A180662 for their definitions, link Catalan's triangle, its mirror is A033184, with several sequences, see crossrefs. - Johannes W. Meijer, Sep 22 2010
The m-th row of Catalan's triangle consists of the unique nonnegative differences of the form binomial(m+k,m)-binomial(m+k,m+1) with 0 <= k <= m (See Links). - R. J. Cano, Jul 22 2014
T(n,k) is also the number of nondecreasing parking functions of length n+1 whose maximum element is k+1. For example T(3,2) = 5 because we have (1,1,1,3), (1,1,2,3), (1,2,2,3), (1,1,3,3), (1,2,3,3). - Ran Pan, Nov 16 2015
T(n,k) is the number of Dyck paths from (0,0) to (n+2,n+2) which start with n-k+2 east steps and touch the diagonal y=x only on the last north step. - Felipe Rueda, Sep 18 2019
T(n-1,k) for k < n is number of well-formed strings of n parenthesis pairs with prefix of exactly n-k opening parenthesis; T(n,n) = T(n,n-1). - Hermann Stamm-Wilbrandt, May 02 2021

			Triangle begins in row n=0 with 0 <= k <= n:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  5,   5;
  1, 4,  9,  14,  14;
  1, 5, 14,  28,  42,   42;
  1, 6, 20,  48,  90,  132,  132;
  1, 7, 27,  75, 165,  297,  429,  429;
  1, 8, 35, 110, 275,  572, 1001, 1430, 1430;
  1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862;

William Feller, Introduction to Probability Theory and its Applications, vol. I, ed. 2, chap. 3, sect. 1,2.
Ki Hang Kim, Douglas G. Rogers, and Fred W. Roush, Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577-594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013).
D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6, Eq. 22, p. 451.
C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146.
M. Bellon, Query 5467, L'Intermédiaire des Mathématiciens, 4 (1925), 11; H. Ory, 4 (1925), 120. - N. J. A. Sloane, Mar 09 2022
Andrzej Proskurowski and Ekaputra Laiman, Fast enumeration, ranking, and unranking of binary trees. Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982). Congr. Numer. 35 (1982), 401-413.MR0725898 (85a:68152).
M. Welsch, Note #371, L'Intermédiaire des Mathématiciens, 2 (1895), pp. 235-237. - N. J. A. Sloane, Mar 02 2022

T. D. Noe, Rows n = 0..100 of triangle, flattened
Erik Aas, Arvind Ayyer, Svante Linusson and Samu Potka, The exact phase diagram for a semipermeable TASEP with nonlocal boundary jumps, arXiv:1902.02019 [cond-mat.stat-mech], 2019.
Ron M. Adin, E. Bagno, and Y. Roichman, Block decomposition of permutations and Schur-positivity, arXiv:1611.06979 [math.CO], 2016-2017.
Kassie Archer, Abigail Bishop, Alexander Diaz-Lopez, Luis David Garcia Puente, Darren Glass, and Joel Louwsma, Arithmetical structures on bidents, arXiv:1903.01393 [math.CO], 2019.
J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.
Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, arXiv:0711.0906 [math.CO], 2007.
Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, Discrete Math., 308 (2008), 4660-4669.
Axel Bacher, Antonio Bernini, Luca Ferrari, Benjamin Gunby, Renzo Pinzani, and Julian West, The Dyck pattern poset Discrete Math. 321 (2014), 12--23. MR3154009.
D. F. Bailey, Counting arrangements of 1's and-1's, Mathematics Magazine, 69 (1996): 128-131. See table on p. 129.
Elena Barcucci, Alberto Del Lungo, Elisa Pergola, and Renzo Pinzani, A methodology for plane tree enumeration, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995). Discrete Math. 180 (1998), no. 1-3, 45--64. MR1603693 (98m:05090).
E. Barcucci and M. C. Verri, Some more properties of Catalan numbers, Discrete Math., 102 (1992), 229-237.
J.-L. Baril, C. Khalil, and V. Vajnovszki, Catalan and Schröder permutations sortable by two restricted stacks, arXiv:2004.01812 [cs.DM], 2020.
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 3.
Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Paul Barry and A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, example 3.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.
A. Bernini, L. Ferrari, R. Pinzani, and J. West, The Dyck pattern poset, arXiv:1303.3785 [math.CO], 2013.
N. Borie, Combinatorics of simple marked mesh patterns in 132-avoiding permutations, arXiv:1311.6292 [math.CO], 2013.
M. Bousquet-Mélou and M. Petkovsek, Linear recurrences with constant coefficients: the multivariate case, Discrete Math. 225 (2000), 51-75.
Jasper Bown, Peter Kagey, Alan Kappler, Michael E. Orrison, and Jayden Thadani, Preference-restricted parking functions, arXiv:2507.11701 [math.CO], 2025. See p. 11.
Benjamin Braun, Hugo Corrales, Scott Corry, Luis David García Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin, Gregg Musiker, and Carlos E. Valencia, Counting Arithmetical Structures on Paths and Cycles. arXiv:1701.06377 [math.CO], 2017.
E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.
S. Brlek, E. Duchi, E. Pergola, and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
Hong Duc Bui, Counting Number of Triangulations of Point Sets: Reinterpreting and Generalizing the Triangulation Polynomials, arXiv:2504.09615 [cs.CG], 2025. See p. 4.
Steve Butler, R. Graham, and C. H. Yan, Parking distributions on trees, European Journal of Combinatorics 65 (2017), 168-185.
R. J. Cano, Catalan's books
L. Carlitz, Sequences, paths, ballot numbers
Douglas M. Chen, On the Structure of Permutation Invariant Parking, arXiv:2311.15699 [math.CO], 2023. See p. 16.
Lapo Cioni and Luca Ferrari, Preimages under the Queuesort algorithm, arXiv preprint arXiv:2102.07628 [math.CO], 2021; Discrete Math., 344 (2021), #112561.
Ari Cruz, Pamela E. Harris, Kimberly J. Harry, Jan Kretschmann, Matt McClinton, Alex Moon, John O. Museus, and Eric Redmon, On some discrete statistics of parking functions, arXiv:2312.16786 [math.CO], 2023.
Italo J. Dejter, A new approach to the middle levels via a Catalan-number system of numeration, 2015.
Italo J. Dejter, A numeral system for the middle levels, preprint, 2014. [See Section 2. - _N. J. A. Sloane_, Apr 06 2014]
Italo J. Dejter, Dihedral-symmetry middle-levels problem via a Catalan system of numeration, preprint, 2015.
Italo J. Dejter, The role of restricted growth strings in the two middle levels of the Boolean lattice B_(2k+1), University of Puerto Rico, 2018.
Italo J. Dejter, Reinterpreting Mütze's Theorem via Natural Enumeration of Ordered Rooted Trees, arXiv:1911.02100 [math.CO], 2019.
Italo J. Dejter, A numeral system for the middle-levels graphs, Elec. J. Graph Theory and Applications (2021) Vol. 9, No. 1, 137-156. See p. 139.
B. Derrida, E. Domany, and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672. (Y_{N}(K) = A009766(N+1,K-1), 1 <= K <= N+1, N >=0 if alpha = 1 = beta).
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
Paul Drube, Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers, arXiv:1606.04869 [math.CO], 2016.
Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Classification of uniform flag triangulations of the Legendre polytope, arXiv:1901.07113 [math.CO], 2019.
R. Ehrenborg, S. Kitaev, and E. Steingrimsson, Number of cycles in the graph of 312-avoiding permutations, arXiv:1310.1520 [math.CO], 2013.
W. J. R. Eplett, A note about the Catalan triangle, Discrete Math. 25(1979), no. 3, 289--291. MR0534947 (80i:05007)
Jackson Evoniuk, Steven Klee, and Van Magnan, Enumerating Minimal Length Lattice Paths, J. Int. Seq., Vol. 21 (2018), Article 18.3.6.
G. Feinberg and K.-H. Lee, Homogeneous representations of KLR-algebras and fully commutative elements, arXiv:1401.0845 [math.RT], 2014.
I. Fanti, A. Frosini, E. Grazzini, R. Pinzani, and S. Rinaldi, Characterization and enumeration of some classes of permutominoes, PU. M. A., Vol. 18 (2007), No. 3-4, pp. 265-290.
Dominique Foata and Guo-Niu Han, The doubloon polynomial triangle, Ram. J. 23 (2010), 107-126.
Dominique Foata and Guo-Niu Han, Doubloons and new q-tangent numbers, Quart. J. Math. 62 (2) (2011) 417-432.
H. G. Forder, Some problems in combinatorics, Math. Gazette, vol. 45, 1961, 199-201.
C. A. Francisco, J. Mermin, and J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, preprint 2013; European Journal of Combinatorics, Volume 45, April 2015, Pages 85-96.
Shishuo Fu, Zhicong Lin, and Yaling Wang, A combinatorial bijection on di-sk trees, arXiv:2011.11302 [math.CO], 2020.
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See p. 12.
Ira Gessel, Super ballot numbers.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Claude Godrèche and Jean-Marc Luck, On the first positive position of a random walker, arXiv:2501.17268 [cond-mat.stat-mech], 2025.
Niket Gowravaram, A Variation of the nil-Temperley-Lieb algebras of type A, Preprint, 2015.
J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.
J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)
Pamela E. Harris, Thiago Holleben, J. Carlos Martínez Mori, Amanda Priestley, Keith Sullivan, and Per Wagenius, A Probabilistic Parking Process and Labeled IDLA, arXiv:2501.11718 [math.PR], 2025. See pp. 8, 28.
David He and Daniel Tubbenhauer, Tensor powers of representations of (diagram) monoids, arXiv:2508.04054 [math.RT], 2025. See p. 7.
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
F. Hivert, J.-C. Novelli, and J.-Y. Thibon, The Algebra of Binary Search Trees, Theoretical Computer Science, 339 (2005), 129-165.
R. L. Hudson and Y. Pei, On a causal quantum stochastic double product integral related to Lévy area, Research Gate, 2015.
R. L. Hudson and Y. Pei, On a quantum causal stochastic double product integral related to Levy area, arXiv:1506.04294 [math-ph], 2015.
Brant Jones, Avoiding patterns and making the best choice, arXiv:1812.00963 [math.CO], 2018.
Cemil Karaçam and Alper Vural, Enumerating 2D and 3D lattice paths with arbitrary steps, Mathematica Bohemica, pp. 1-13 (2025). See p. 12.
Antii Karttunen, Some notes on Catalan's Triangle.
Lord C. Kavi and Michael W. Newman, Counting closed walks in infinite regular trees using Catalan and Borel's triangles, arXiv:2212.08795 [math.CO], 2022.
Dongsu Kim and Zhicong Lin, Refined restricted inversion sequences, arXiv:1706.07208 [math.CO], 2017.
W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146. [Annotated scanned copy]
A. Laradji and A. Umar, On certain finite semigroups of order-decreasing transformations I, Semigroup Forum 69 (2004), 184-200.
Jeong-Yup Lee, Dong-il Lee, and Sungsoon Kim, Gröbner-Shirshov bases for Temperley-Lieb algebras of the complex reflection group of type G(d,1,n), arXiv:1808.06523 [math.RA], 2018.
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Shao-Hua Liu, The operators F_i on permutations, 132-avoiding permutations and inversions, Discrete Math., 342 (2019), 2402-2414.
Jiaxi Lu and Yuanzhe Ding, A skeleton model to enumerate standard puzzle sequences, arXiv:2106.09471 [math.CO], 2021.
D. Merlini et al., Underdiagonal lattice paths with unrestricted steps, Discrete Appl. Math., 91 (1999), 197-213.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table I).
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv:1508.03757 [math.RA], 2015.
J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.
M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, and C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014 and J. Int. Seq. 18 (2015) 15.10.6.
P. Pagacz and M. Wojtylak, On the spectral properties of a class of H-selfadjoint random matrices and the underlying combinatorics, arXiv:1310.2122 [math.PR], 2013.
R. Parviainen, Permutations, cycles and the pattern 2-13, arXiv:math/0607793 [math.CO], 2006.
R. Pemantle and M. C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, SIAM Rev., 50 (2008), no. 2, 199-272. See p. 261
Planet Math, Lattice Paths and ballot numbers.
L. Pudwell, Avoiding an Ordered Partition of Length 3, 2013.
C. M. Ringel, The Catalan combinatorics of the hereditary artin algebras, arXiv:1502.06553 [math.RT], 2015.
Gabriel Bravo Rios and Agustin Moreno Cañadas, Dyck Paths in Representation Theory of Algebras, National University of Colombia (2020).
A. Robertson, D. Saracino, and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
L. W. Shapiro, A Catalan triangle, Discrete Math., 14, 83-90, 1976.
T. Sillke, Catalan's numbers
Hermann Stamm-Wilbrandt, Visualization for all well-formed strings of 4 parenthesis pairs and relation to T(n,k).
R. A. Sulanke, Guessing, ballot numbers and refining Pascal's triangle
Yidong Sun, A simple bijection between binary trees and colored ternary trees, El. J. Combinat. 17 (2010) #N20
Paweł J. Szabłowski, Yet another way of calculating moments of the Kesten's distribution and its consequences, arXiv:2106.10461 [math.CO], 2021.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024. See pp. 9, 29.
Rafael Vazquez and M. Krstic, Boundary control of a singular reaction-diffusion equation on a disk, arXiv:1601.02010 [math.OC], 2016.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Eric Weisstein's World of Mathematics, Catalan's Triangle
Eric Weisstein's World of Mathematics, Nonnegative Partial Sum
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 17.
Martha Yip, Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices, arXiv:1703.00057 [math.CO], 2017; The Electronic Journal of Combinatorics 25(1) (2018), #P1.68.

The following are all versions of (essentially) the same Catalan triangle: A009766, A008315, A028364, A030237, A047072, A053121, A059365, A062103, A099039, A106566, A130020, A140344.
Cf. A062745, A214292.
Sums of the shallow diagonals give A001405, central binomial coefficients: 1=1, 1=1, 1+1=2, 1+2=3, 1+3+2=6, 1+4+5=10, 1+5+9+5=20, ...
Row sums as well as the sums of the squares of the shallow diagonals give Catalan numbers (A000108).
Reflected version of A033184.
Diagonals give A000108, A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...
Triangle sums (see the comments): A000108 (Row1), A000957 (Row2), A001405 (Kn11), A014495 (Kn12), A194124 (Kn13), A030238 (Kn21), A000984 (Kn4), A000958 (Fi2), A165407 (Ca1), A026726 (Ca4), A081696 (Ze2).

Flat(List([0..10],n->List([0..n],m->Binomial(n+m,n)*(n-m+1)/(n+1)))); # Muniru A Asiru, Feb 18 2018

a009766 n k = a009766_tabl !! n !! k
a009766_row n = a009766_tabl !! n
a009766_tabl = iterate (\row -> scanl1 (+) (row ++ [0])) [1]
-- Reinhard Zumkeller, Jul 12 2012

[[Binomial(n+k,n)*(n-k+1)/(n+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 07 2019

A009766 := proc(n,k) binomial(n+k,n)*(n-k+1)/(n+1); end proc:
seq(seq(A009766(n,k), k=0..n), n=0..10); # R. J. Mathar, Dec 03 2010

Flatten[NestList[Append[Accumulate[#], Last[Accumulate[#]]] &, {1}, 9]] (* Birkas Gyorgy, May 19 2012 *)
T[n_, k_] := T[n, k] = Which[k == 0, 1, k>n, 0, True, T[n-1, k] + T[n, k-1] ]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2016 *)

{T(n, k) = if( k<0 || k>n, 0, binomial(n+1+k, k) * (n+1-k) / (n+1+k) )}; /* Michael Somos, Oct 17 2006 */

b009766=(n1=0,n2=100)->{my(q=if(!n1,0,binomial(n1+1,2)));for(w=n1,n2,for(k=0,w,write("b009766.txt",1*q" "1*(binomial(w+k,w)-binomial(w+k,w+1)));q++))} \\ R. J. Cano, Jul 22 2014

from math import comb, isqrt
def A009766(n): return comb((a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))+(b:=n-comb(a+1,2)),b)*(a-b+1)//(a+1) # Chai Wah Wu, Nov 12 2024

@CachedFunction
def ballot(p,q):
    if p == 0 and q == 0: return 1
    if p < 0 or p > q: return 0
    S = ballot(p-2, q) + ballot(p, q-2)
    if q % 2 == 1: S += ballot(p-1, q-1)
    return S
A009766 = lambda n, k: ballot(2*k, 2*n)
for n in (0..7): [A009766(n, k) for k in (0..n)]  # Peter Luschny, Mar 05 2014

[[binomial(n+k,n)*(n-k+1)/(n+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 07 2019

T(n, m) = binomial(n+m, n)*(n-m+1)/(n+1), 0 <= m <= n.
G.f. for column m: (x^m)*N(2; m-1, x)/(1-x)^(m+1), m >= 0, with the row polynomials from triangle A062991 and N(2; -1, x) := 1.
G.f.: C(t*x)/(1-x*C(t*x)) = 1 + (1+t)*x + (1+2*t+2*t^2)*x^2 + ..., where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function. - Emeric Deutsch, May 18 2004
Another version of triangle T(n, k) given by [1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 1, 1, 1, 1, 1, ...] = 1; 1, 0; 1, 1, 0; 1, 2, 2, 0; 1, 3, 5, 5, 0; 1, 4, 9, 14, 14, 0; ... where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 16 2005
O.g.f. (with offset 1) is the series reversion of x*(1+x*(1-t))/(1+x)^2 = x - x^2*(1+t) + x^3*(1+2*t) - x^4*(1+3*t) + ... . - Peter Bala, Jul 15 2012
Sum_{k=0..floor(n/2)} T(n-k+p-1, k+p-1) = A001405(n+2*p-2) - C(n+2*p-2, p-2), p >= 1. - Johannes W. Meijer, Oct 03 2013
Let A(x,t) denote the o.g.f. Then 1 + x*d/dx(A(x,t))/A(x,t) = 1 + (1 + t)*x + (1 + 2*t + 3*t^2)*x^2 + (1 + 3*t + 6*t^2 + 10*t^3)*x^3 + ... is the o.g.f. for A059481. - Peter Bala, Jul 21 2015
The n-th row polynomial equals the n-th degree Taylor polynomial of the function (1 - 2*x)/(1 - x)^(n+2) about 0. For example, for n = 4, (1 - 2*x)/(1 - x)^6 = 1 + 4*x + 9*x^2 + 14*x^3 + 14*x^4 + O(x^5). - Peter Bala, Feb 18 2018
T(n,k) = binomial(n + k + 1, k) - 2*binomial(n + k, k - 1), for 0 <= k <= n. - David Callan, Jun 15 2022

A005258 Apéry numbers: a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,k).

Original entry on oeis.org

1, 3, 19, 147, 1251, 11253, 104959, 1004307, 9793891, 96918753, 970336269, 9807518757, 99912156111, 1024622952993, 10567623342519, 109527728400147, 1140076177397091, 11911997404064793, 124879633548031009, 1313106114867738897, 13844511065506477501

Offset: 0

N. J. A. Sloane

This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004
Equals the main diagonal of square array A108625. - Paul D. Hanna, Jun 14 2005
This sequence is t_5 in Cooper's paper. - Jason Kimberley, Nov 25 2012
Conjecture: For each n=1,2,3,... the polynomial a_n(x) = Sum_{k=0..n} C(n,k)^2*C(n+k,k)*x^k is irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 21 2013
Diagonal of rational functions 1/(1 - x - x*y - y*z - x*z - x*y*z), 1/(1 + y + z + x*y + y*z + x*z + x*y*z), 1/(1 - x - y - z + x*y + x*y*z), 1/(1 - x - y - z + y*z + x*z - x*y*z). - Gheorghe Coserea, Jul 07 2018

			G.f. = 1 + 3*x + 19*x^2 + 147*x^3 + 1251*x^4 + 11253*x^5 + 104959*x^6 + ...

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
S. Melczer, An Invitation to Analytic Combinatorics, 2021; p. 129.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Simon Plouffe, Table of n, a(n) for n = 0..954
B. Adamczewski, J. P. Bell, and E. Delaygue, Algebraic independence of G-functions and congruences "a la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
Roger Apéry, Irrationalité de zeta(2) et zeta(3), in Journées Arith. de Luminy. Colloque International du Centre National de la Recherche Scientifique (CNRS) held at the Centre Universitaire de Luminy, Luminy, Jun 20-24, 1978. Astérisque, 61 (1979), 11-13.
Roger Apéry, Sur certaines séries entières arithmétiques, Groupe de travail d'analyse ultramétrique, 9 no. 1 (1981-1982), Exp. No. 16, 2 p.
Thomas Baruchel and C. Elsner, On error sums formed by rational approximations with split denominators, arXiv preprint arXiv:1602.06445 [math.NT], 2016.
Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982, page 657.
F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201-210.
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Francis Brown, Irrationality proofs for zeta values, moduli spaces and dinner parties, arXiv:1412.6508 [math.NT], 2014.
Shaun Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.
M. Coster, Email, Nov 1990
E. Delaygue, Arithmetic properties of Apéry-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013-2015.
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Number Theory 117 (2006), 191-215.
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
C. Elsner, On prime-detecting sequences from Apéry's recurrence formulas for zeta(3) and zeta(2), JIS 11 (2008) 08.5.1.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See D p. 2.
R. K. Guy, Letter to N. J. A. Sloane, Oct 1985
S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.
Michael D. Hirschhorn, A Connection Between Pi and Phi, Fibonacci Quart. 53 (2015), no. 1, 42-47.
Lalit Jain and Pavlos Tzermias, Beukers' integrals and Apéry's recurrences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.1.
Bradley Klee, Checking Weierstrass data, 2023.
Vaclav Kotesovec, Asymptotic of generalized Apéry sequences with powers of binomial coefficients, Nov 04 2012.
Ji-Cai Liu, Supercongruences for the (p-1)th Apéry number, arXiv:1803.11442 [math.NT], 2018.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Peter Paule and Carsten Schneider, Computer proofs of a new family of harmonic number identities, Advances in Applied Mathematics (31), 359-378, (2003).
Simon Plouffe, The first 2553 Apéry numbers
E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635 [math.NT], 2013.
V. Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.
A. van der Poorten, A proof that Euler missed ... Apéry's proof of the irrationality of zeta(3). An informal report. Math. Intelligencer 1 (1978/79), no 4, 195-203.
Eric Weisstein's World of Mathematics, Apéry Number.
D. Zagier, Integral solutions of Apéry-like recurrence equations. See line D in sporadic solutions table of page 5.
W. Zudilin, Approximations to -, di- and tri-logarithms, arXiv:math/0409023 [math.CA], 2004-2005.

Cf. A002736, A005259, A005429, A005430, A108625, A143413, A218690, A218692.
Cf. A007318.
Cf. A001008, A002805.
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

a:=n->Sum([0..n],k->(-1)^(n-k)*Binomial(n,k)*Binomial(n+k,k)^2);;
A005258:=List([0..20],n->a(n));; # Muniru A Asiru, Feb 11 2018

List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*Binomial(n+k,k))); # Muniru A Asiru, Jul 29 2018

a005258 n = sum [a007318 n k ^ 2 * a007318 (n + k) k | k <- [0..n]]
-- Reinhard Zumkeller, Jan 04 2013

[&+[Binomial(n,k)^2 * Binomial(n+k,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Nov 28 2018

with(combinat): seq(add((multinomial(n+k,n-k,k,k))*binomial(n,k), k=0..n), n=0..18); # Zerinvary Lajos, Oct 18 2006
a := n -> binomial(2*n, n)*hypergeom([-n, -n, -n], [1, -2*n], 1):
seq(simplify(a(n)), n=0..20); # Peter Luschny, Feb 10 2018

a[n_] := HypergeometricPFQ[ {n+1, -n, -n}, {1, 1}, 1]; Table[ a[n], {n, 0, 18}] (* Jean-François Alcover, Jan 20 2012, after Vladeta Jovovic *)
Table[Sum[Binomial[n,k]^2 Binomial[n+k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Aug 25 2019 *)

{a(n) = if( n<0, -(-1)^n * a(-1-n), sum(k=0, n, binomial(n, k)^2 * binomial(n+k, k)))} /* Michael Somos, Sep 18 2013 */

def A005258(n):
    m, g = 1, 0
    for k in range(n+1):
        g += m
        m *= (n+k+1)*(n-k)**2
        m //= (k+1)**3
    return g # Chai Wah Wu, Oct 02 2022

a(n) = hypergeom([n+1, -n, -n], [1, 1], 1). - Vladeta Jovovic, Apr 24 2003
D-finite with recurrence: (n+1)^2 * a(n+1) = (11*n^2+11*n+3) * a(n) + n^2 * a(n-1). - Matthijs Coster, Apr 28 2004
Let b(n) be the solution to the above recurrence with b(0) = 0, b(1) = 5. Then the b(n) are rational numbers with b(n)/a(n) -> zeta(2) very rapidly. The identity b(n)*a(n-1) - b(n-1)*a(n) = (-1)^(n-1)*5/n^2 leads to a series acceleration formula: zeta(2) = 5 * Sum_{n >= 1} 1/(n^2*a(n)*a(n-1)) = 5*(1/(1*3) + 1/(2^2*3*19) + 1/(3^2*19*147) + ...). Similar results hold for the constant e: see A143413. - Peter Bala, Aug 14 2008
G.f.: hypergeom([1/12, 5/12],[1], 1728*x^5*(1-11*x-x^2)/(1-12*x+14*x^2+12*x^3+x^4)^3) / (1-12*x+14*x^2+12*x^3+x^4)^(1/4). - Mark van Hoeij, Oct 25 2011
a(n) ~ ((11+5*sqrt(5))/2)^(n+1/2)/(2*Pi*5^(1/4)*n). - Vaclav Kotesovec, Oct 05 2012
1/Pi = 5*(sqrt(47)/7614)*Sum_{n>=0} (-1)^n a(n)*binomial(2n,n)*(682n+71)/15228^n. [Cooper, equation (4)] - Jason Kimberley, Nov 26 2012
a(-1 - n) = (-1)^n * a(n) if n>=0. a(-1 - n) = -(-1)^n * a(n) if n<0. - Michael Somos, Sep 18 2013
0 = a(n)*(a(n+1)*(+4*a(n+2) + 83*a(n+3) - 12*a(n+4)) + a(n+2)*(+32*a(n+2) + 902*a(n+3) - 147*a(n+4)) + a(n+3)*(-56*a(n+3) + 12*a(n+4))) + a(n+1)*(a(n+1)*(+17*a(n+2) + 374*a(n+3) - 56*a(n+4)) + a(n+2)*(+176*a(n+2) + 5324*a(n+3) - 902*a(n+4)) + a(n+3)*(-374*a(n+3) + 83*a(n+4))) + a(n+2)*(a(n+2)*(-5*a(n+2) - 176*a(n+3) + 32*a(n+4)) + a(n+3)*(+17*a(n+3) - 4*a(n+4))) for all n in Z. - Michael Somos, Aug 06 2016
a(n) = binomial(2*n, n)*hypergeom([-n, -n, -n],[1, -2*n], 1). - Peter Luschny, Feb 10 2018
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*binomial(n+k,k)^2. - Peter Bala, Feb 10 2018
G.f. y=A(x) satisfies: 0 = x*(x^2 + 11*x - 1)*y'' + (3*x^2 + 22*x - 1)*y' + (x + 3)*y. - Gheorghe Coserea, Jul 01 2018
From Peter Bala, Jan 15 2020: (Start)
a(n) = Sum_{0 <= j, k <= n} (-1)^(j+k)*C(n,k)*C(n+k,k)^2*C(n,j)* C(n+k+j,k+j).
a(n) = Sum_{0 <= j, k <= n} (-1)^(n+j)*C(n,k)^2*C(n+k,k)*C(n,j)* C(n+k+j,k+j).
a(n) = Sum_{0 <= j, k <= n} (-1)^j*C(n,k)^2*C(n,j)*C(3*n-j-k,2*n). (End)
a(n) = [x^n] 1/(1 - x)*( Legendre_P(n,(1 + x)/(1 - x)) )^m at m = 1. At m = 2 we get the Apéry numbers A005259. - Peter Bala, Dec 22 2020
a(n) = (-1)^n*Sum_{j=0..n} (1 - 5*j*H(j) + 5*j*H(n - j))*binomial(n, j)^5, where H(n) denotes the n-th harmonic number, A001008/A002805. (Paule/Schneider). - Peter Luschny, Jul 23 2021
From Bradley Klee, Jun 05 2023: (Start)
The g.f. T(x) obeys a period-annihilating ODE:
0=(3 + x)*T(x) + (-1 + 22*x + 3*x^2)*T'(x) + x*(-1 + 11*x + x^2)*T''(x).
The periods ODE can be derived from the following Weierstrass data:
g2 = 3*(1 - 12*x + 14*x^2 + 12*x^3 + x^4);
g3 = 1 - 18*x + 75*x^2 + 75*x^4 + 18*x^5 + x^6;
which determine an elliptic surface with four singular fibers. (End)
Conjecture: a(n)^2 = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*A143007(n, k). - Peter Bala, Jul 08 2024

A027383 a(2n) = 32^n - 2; a(2*n+1) = 2^(n+2) - 2.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 22, 30, 46, 62, 94, 126, 190, 254, 382, 510, 766, 1022, 1534, 2046, 3070, 4094, 6142, 8190, 12286, 16382, 24574, 32766, 49150, 65534, 98302, 131070, 196606, 262142, 393214, 524286, 786430, 1048574, 1572862, 2097150, 3145726, 4194302, 6291454

Offset: 0

R. K. Guy

Number of balanced strings of length n: let d(S) = #(1's) - #(0's), # == count in S, then S is balanced if every substring T of S has -2 <= d(T) <= 2.
Number of "fold lines" seen when a rectangular piece of paper is folded n+1 times along alternate orthogonal directions and then unfolded. - Quim Castellsaguer (qcastell(AT)pie.xtec.es), Dec 30 1999
Also the number of binary strings with the property that, when scanning from left to right, once the first 1 is seen in position j, there must be a 1 in positions j+2, j+4, ... until the end of the string. (Positions j+1, j+3, ... can be occupied by 0 or 1.) - Jeffrey Shallit, Sep 02 2002
a(n-1) is also the Moore lower bound on the order of a (3,n)-cage. - Eric W. Weisstein, May 20 2003 and Jason Kimberley, Oct 30 2011
Partial sums of A016116. - Hieronymus Fischer, Sep 15 2007
Equals row sums of triangle A152201. - Gary W. Adamson, Nov 29 2008
From John P. McSorley, Sep 28 2010: (Start)
a(n) = DPE(n+1) is the total number of k-double-palindromes of n up to cyclic equivalence. See sequence A180918 for the definitions of a k-double-palindrome of n and of cyclic equivalence. Sequence A180918 is the 'DPE(n,k)' triangle read by rows where DPE(n,k) is the number of k-double-palindromes of n up to cyclic equivalence. For example, we have a(4) = DPE(5) = DPE(5,1) + DPE(5,2) + DPE(5,3) + DPE(5,4) + DPE(5,5) = 0 + 2 + 2 + 1 + 1 = 6.
The 6 double-palindromes of 5 up to cyclic equivalence are 14, 23, 113, 122, 1112, 11111. They come from cyclic equivalence classes {14,41}, {23,32}, {113,311,131}, {122,212,221}, {1112,2111,1211,1121}, and {11111}. Hence a(n)=DPE(n+1) is the total number of cyclic equivalence classes of n containing at least one double-palindrome.
(End)
From Herbert Eberle, Oct 02 2015: (Start)
For n > 0, there is a red-black tree of height n with a(n-1) internal nodes and none with less.
In order a red-black tree of given height has minimal number of nodes, it has exactly 1 path with strictly alternating red and black nodes. All nodes outside this height defining path are black.
Consider:
mrbt5                R
                    / \
                   /   \
                  /     \
                 /       B
                /       / \
mrbt4          B       /   B
              / \     B   E E
             /   B   E E
mrbt3       R   E E
           / \
          /   B
mrbt2    B   E E
        / E
mrbt1  R
      E E
(Red nodes shown as R, blacks as B, externals as E.)
Red-black trees mrbt1, mrbt2, mrbt3, mrbt4, mrbt5 of respective heights h = 1, 2, 3, 4, 5; all minimal in the number of internal nodes, namely 1, 2, 4, 6, 10.
Recursion (let n = h-1): a(-1) = 0, a(n) = a(n-1) + 2^floor(n/2), n >= 0.
(End)
Also the number of strings of length n with the digits 1 and 2 with the property that the sum of the digits of all substrings of uneven length is not divisible by 3. An example with length 8 is 21221121. - Herbert Kociemba, Apr 29 2017
a(n-2) is the number of achiral n-bead necklaces or bracelets using exactly two colors.  For n=4, the four arrangements are AAAB, AABB, ABAB, and ABBB. - Robert A. Russell, Sep 26 2018
Partial sums of powers of 2 repeated 2 times, like A200672 where is 3 times. - Yuchun Ji, Nov 16 2018
Also the number of binary words of length n with cuts-resistance <= 2, where, for the operation of shortening all runs by one, cuts-resistance is the number of applications required to reach an empty word. Explicitly, these are words whose sequence of run-lengths, all of which are 1 or 2, has no odd-length run of 1's sandwiched between two 2's. - Gus Wiseman, Nov 28 2019
Also the number of up-down paths with n steps such that the height difference between the highest and lowest points is at most 2. - Jeremy Dover, Jun 17 2020
Also the number of non-singleton integer compositions of n + 2 with no odd part other than the first or last. Including singletons gives A052955. This is an unsorted (or ordered) version of A351003. The version without even (instead of odd) interior parts is A001911, complement A232580. Note that A000045(n-1) counts compositions without odd parts, with non-singleton case A077896, and A052952/A074331 count non-singleton compositions without even parts. Also the number of compositions y of n + 1 such that y_i = y_{i+1} for all even i. - Gus Wiseman, Feb 19 2022

			After 3 folds one sees 4 fold lines.
Example: a(3) = 6 because the strings 001, 010, 100, 011, 101, 110 have the property.
Binary: 1, 10, 100, 110, 1010, 1110, 10110, 11110, 101110, 111110, 1011110, 1111110, 10111110, 11111110, 101111110, 111111110, 1011111110, 1111111110, 10111111110, ... - _Jason Kimberley_, Nov 02 2011
Example: Partial sums of powers of 2 repeated 2 times:
a(3) = 1+1+2 = 4;
a(4) = 1+1+2+2 = 6;
a(5) = 1+1+2+2+4 = 10.
_Yuchun Ji_, Nov 16 2018

John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010. [John P. McSorley, Sep 28 2010]

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. - From _N. J. A. Sloane_, Feb 03 2013
Leonard F. Klosinski, Gerald L. Alexanderson and Loren C. Larson, Under misprinted head B3, Amer Math. Monthly, 104(1997) 753-754.
Laurent Noé, Spaced seed design on profile HMMs for precise HTS read-mapping efficient sliding window product on the matrix semi-group, in Rapide Bilan 2012-2013, LIFL, Université Lille 1 - INRIA Journées au vert 11 et 12 juin 2013.
Eric Weisstein's World of Mathematics, Cage Graph
Index entries for sequences obtained by enumerating foldings
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A132666, A152201.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), this sequence (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A000066 (actual order of a (3,g)-cage).
Bisections are A033484 (even) and A000918 (odd).
a(n) = A305540(n+2,2), the second column of the triangle.
Numbers whose binary expansion is a balanced word are A330029.
Binary words counted by cuts-resistance are A319421 or A329860.
Cf. A000975, A062011, A329862, A329863.
The complementary compositions are counted by A274230(n-1) + 1, with bisections A060867 (even) and A134057 (odd).
Cf. A000346, A000984, A001405, A001700, A011782 (compositions).
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

import Data.List (transpose)
a027383 n = a027383_list !! n
a027383_list = concat $ transpose [a033484_list, drop 2 a000918_list]
-- Reinhard Zumkeller, Jun 17 2015

[2^Floor((n+2)/2)+2^Floor((n+1)/2)-2: n in [0..50]]; // Vincenzo Librandi, Aug 16 2011

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=2*a[n-2]+2 od: seq(a[n], n=1..41); # Zerinvary Lajos, Mar 16 2008

a[n_?EvenQ] := 3*2^(n/2)-2; a[n_?OddQ] := 2^(2+(n-1)/2)-2; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 21 2011, after Quim Castellsaguer *)
LinearRecurrence[{1, 2, -2}, {1, 2, 4}, 41] (* Robert G. Wilson v, Oct 06 2014 *)
Table[Length[Select[Tuples[{0,1},n],And[Max@@Length/@Split[#]<=2,!MatchQ[Length/@Split[#],{_,2,ins:1..,2,_}/;OddQ[Plus[ins]]]]&]],{n,0,15}] (* Gus Wiseman, Nov 28 2019 *)

a(n)=2^(n\2+1)+2^((n+1)\2)-2 \\ Charles R Greathouse IV, Oct 21 2011

def a(n): return 2**((n+2)//2) + 2**((n+1)//2) - 2
print([a(n) for n in range(43)]) # Michael S. Branicky, Feb 19 2022

a(0)=1, a(1)=2; thereafter a(n+2) = 2*a(n) + 2.
a(2n) = 3*2^n - 2 = A033484(n);
a(2n-1) = 2^(n+1) - 2 = A000918(n+1).
G.f.: (1 + x)/((1 - x)*(1 - 2*x^2)). - David Callan, Jul 22 2008
a(n) = Sum_{k=0..n} 2^min(k, n-k).
a(n) = 2^floor((n+2)/2) + 2^floor((n+1)/2) - 2. - Quim Castellsaguer (qcastell(AT)pie.xtec.es)
a(n) = 2^(n/2)*(3 + 2*sqrt(2) + (3-2*sqrt(2))*(-1)^n)/2 - 2. - Paul Barry, Apr 23 2004
a(n) = A132340(A052955(n)). - Reinhard Zumkeller, Aug 20 2007
a(n) = A052955(n+1) - 1. - Hieronymus Fischer, Sep 15 2007
a(n) = A132666(a(n+1)) - 1. - Hieronymus Fischer, Sep 15 2007
a(n) = A132666(a(n-1)+1) for n > 0. - Hieronymus Fischer, Sep 15 2007
A132666(a(n)) = a(n-1) + 1 for n > 0. - Hieronymus Fischer, Sep 15 2007
G.f.: (1 + x)/((1 - x)*(1 - 2*x^2)). - David Callan, Jul 22 2008
a(n) = 2*( (a(n-2)+1) mod (a(n-1)+1) ), n > 1. - Pierre Charland, Dec 12 2010
a(n) = A136252(n-1) + 1, for n > 0. - Jason Kimberley, Nov 01 2011
G.f.: (1+x*R(0))/(1-x), where R(k) = 1 + 2*x/( 1 - x/(x + 1/R(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013
a(n) = 2^((2*n + 3*(1-(-1)^n))/4)*3^((1+(-1)^n)/2) - 2. - Luce ETIENNE, Sep 01 2014
a(n) = a(n-1) + 2^floor((n-1)/2) for n>0, a(0)=1. - Yuchun Ji, Nov 23 2018
E.g.f.: 3*cosh(sqrt(2)*x) - 2*cosh(x) + 2*sqrt(2)*sinh(sqrt(2)*x) - 2*sinh(x). - Stefano Spezia, Apr 06 2022

More terms from Larry Reeves (larryr(AT)acm.org), Mar 24 2000

Replaced definition with a simpler one. - N. J. A. Sloane, Jul 09 2022

A006480 De Bruijn's S(3,n): (3n)!/(n!)^3.

Original entry on oeis.org

1, 6, 90, 1680, 34650, 756756, 17153136, 399072960, 9465511770, 227873431500, 5550996791340, 136526995463040, 3384731762521200, 84478098072866400, 2120572665910728000, 53494979785374631680, 1355345464406015082330, 34469858696831179429500, 879619727485803060256500, 22514366432046593564460000

Offset: 0

N. J. A. Sloane

Number of paths of length 3n in an n X n X n grid from (0,0,0) to (n,n,n), using steps (0,0,1), (0,1,0), and (1,0,0).
Appears in Ramanujan's theory of elliptic functions of signature 3.
S(s,n) = Sum_{k=0..2n} (-1)^(k+n) * binomial(2n, k)^s. The formula S(3,n) = (3n)!/(n!)^3 is due to Dixon (according to W. N. Bailey 1935). - Charles R Greathouse IV, Dec 28 2011
a(n) is the number of ballot results that end in a 3-way tie when 3n voters each cast two votes for two out of three candidates vying for 2 slots on a county board; in such a tie, each of the three candidates receives 2n votes. Note there are C(3n,2n) ways to choose the voters who cast a vote for the youngest candidate. The n voters who did note vote for the youngest candidate voted for the two older candidates. Then there are C(2n,n) ways to choose the other n voters who voted for both the youngest and the second youngest candidate. The remaining voters vote for the oldest candidate. Hence there are C(3n,2n)*C(2n,n)=(3n)!/(n!)^3 ballot results. - Dennis P. Walsh, May 02 2013
a(n) is the constant term of (X+Y+1/(X*Y))^(3*n). - Mark van Hoeij, May 07 2013
For n > 2 a(n) is divisible by (n+2)*(n+1)^2, a(n) = (n+1)^2*(n+2)*A161581(n). - Alexander Adamchuk, Dec 27 2013
a(n) is the number of permutations of the multiset {1^n, 2^n, 3^n}, the number of ternary words of length 3*n with n of each letters. - Joerg Arndt, Feb 28 2016
Diagonal of the rational function 1/(1 - x - y - z). - Gheorghe Coserea, Jul 06 2016
At least two families of elliptic curves, x = 2*H1 = (p^2+q^2)*(1-q) and x = 2*H2 = p^2+q^2-3*p^2*q+q^3 (0Bradley Klee, Feb 25 2018
The ordinary generating function also determines periods along a family of tetrahedral-symmetric sphere curves ("du troisième ordre"). Compare links to Goursat "Étude des surfaces..." and "Proof Certificate". - Bradley Klee, Sep 28 2018

			G.f.: 1 + 6*x + 90*x^2 + 1680*x^3 + 34650*x^4 + 756756*x^5 + 17153136*x^6 + ...

L. A. Aizenberg and A. P. Yuzhakov, "Integral representations and residues in multidimensional complex analysis", American Mathematical Society, 1983, p. 194.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 174.
N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapters 4 and 6.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
George E. Andrews, The well-poised thread: An Organized Chronicle of Some Amazing Summations and their Implications, Ramanujan J., 1 (1997), 7-23; see Section 8.
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, Journal of Physics A: Mathematical and Theoretical, Vol. 48, No. 50 (2015), 504001; arXiv preprint, arXiv:1507.03227 [math-ph], 2015.
Alin Bostan, Armin Straub, and Sergey Yurkevich, On the representability of sequences as constant terms, arXiv:2212.10116 [math.NT], 2022.
Harlan J. Brothers, Pascal's Prism: Supplementary Material.
Robert M. Dickau, 3-dimensional shortest-path diagrams.
Henry W. Gould, Tables of Combinatorial Identities, Edited by J. Quaintance.
Édouard Goursat, Étude des surfaces qui admettent tous les plans de symétrie d'un polyèdre régulier, Annales scientifiques de l'École Normale Supérieure, Série 3 : Volume 4 (1887), 165-166.
Bob Hinman, Letter to N. J. A. Sloane, Aug. 1980.
Brad Klee, Geometric G.F. for Ramanujan Periods, seqfans mailing list, 2017.
Bradley Klee, Proof Certificate.
Markus Kuba and Alois Panholzer, Lattice paths and the diagonal of the cube, arXiv:2411.03930 [math.CO], 2024.
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavík, Iceland DMTCS proc. AO, 2011, pp. 599-610.
Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, Discrete Mathematics, Vol. 340, No. 10 (2017), pp. 2388-2397; arXiv preprint, arXiv:1602.04347 [math.NT], 2016.
Jovan Mikić, A Method For Examining Divisibility Properties Of Some Binomial Sums, J. Int. Seq., Vol. 21 (2018), Article 18.8.7.
Michaël Moortgat, The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus, 15th Workshop: Computational Logic and Applications (CLA 2020).
Trey Peck, Letter to N. J. A. Sloane, Aug. 1980.
Karol A. Penson and Allan I. Solomon, Coherent states from combinatorial sequences, in: E. Kapuscik and A. Horzela (eds.), Quantum theory and symmetries, World Scientific, 2002, pp. 527-530; arXiv preprint, arXiv:quant-ph/0111151, 2001.
Marko Petkovsek, Herbert Wilf and Doron Zeilberger, A=B, A K Peters, 1996, p. 22.
Srinivasa Ramanujan, Modular Equations and Approximations to Pi, Quarterly Journal of Mathematics, XLV (1914), 350-372.
Bruno Salvy, GFUN and the AGM.
Renzo Sprugnoli, Riordan array proofs of identities in Gould's book, 2006.
Dennis P. Walsh, The probability of a run-off election when three equally-favored candidates vie for two slots.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups".
Eric Weisstein's World of Mathematics, Binomial Sums.
Wikipedia, Dixon's identity.

Cf. A000984, A001459, A008977, A050983, A050984, A091683, A161581, A181545.
Row 3 of A187783.
Related to diagonal of rational functions: A268545-A268555. Elliptic Integrals: A002894, A113424, A000897. Factors: A005809, A000984. Integrals: A007004, A024486. Sphere Curves: A318245, A318495.

List([0..20],n->Factorial(3*n)/Factorial(n)^3); # Muniru A Asiru, Mar 31 2018

[Factorial(3*n)/(Factorial(n))^3: n in [0..20] ]; // Vincenzo Librandi, Aug 20 2011

seq((3*n)!/(n!)^3, n=0..16); # Zerinvary Lajos, Jun 28 2007

Sum [ (-1)^(k+n) Binomial[ 2n, k ]^3, {k, 0, 2n} ]
a[ n_] := If[ n < 0, 0, (-1)^n HypergeometricPFQ[ {-2 n, -2 n, -2 n}, {1, 1}, 1]]; (* Michael Somos, Oct 22 2014 *)
Table[Multinomial[n, n, n], {n, 0, 100}] (* Emanuele Munarini, Oct 25 2016 *)
CoefficientList[Series[Hypergeometric2F1[1/3,2/3,1,27*x],{x,0,5}],x] (* Bradley Klee, Feb 28 2018 *)
Table[(3n)!/(n!)^3,{n,0,20}] (* Harvey P. Dale, Mar 09 2025 *)

makelist(multinomial_coeff(n,n,n),n,0,24); /* Emanuele Munarini, Oct 25 2016 */

{a(n) = if( n<0, 0, (3*n)! / n!^3)}; /* Michael Somos, Dec 03 2002 */

{a(n) = my(A, m); if( n<1, n==0, m=1; A = 1 + O(x); while( m<=n, m*=3; A = subst( (1 + 2*x) * subst(A, x, (x/3)^3), x, serreverse(x * (1 + x + x^2) / (1 + 2*x)^3 / 3 + O(x^m)))); polcoeff(A, n))}; /* Michael Somos, Dec 03 2002 */

from math import factorial
def A006480(n): return factorial(3*n)//factorial(n)**3 # Chai Wah Wu, Oct 04 2022

Using Stirling's formula in A000142 it is easy to get the asymptotic expression a(n) ~ 1/2 * sqrt(3) * 27^n / (Pi*n) - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
From Karol A. Penson, Nov 21 2001: (Start)
O.g.f.: hypergeom([1/3, 2/3], [1], 27*x).
E.g.f.: hypergeom([1/3, 2/3], [1, 1], 27*x).
Integral representation as n-th moment of a positive function on [0, 27]:
a(n) = int( x^n*(-1/24*(3*sqrt(3)*hypergeom([2/3, 2/3], [4/3], 1/27*x)* Gamma(2/3)^6*x^(1/3) - 8*hypergeom([1/3, 1/3], [2/3], 1/27*x)*Pi^3)/Pi^3 /x^(2/3)/Gamma(2/3)^3), x=0..27). This representation is unique. (End)
a(n) = Sum_{k=-n..n} (-1)^k*binomial(2*n, n+k)^3. - Benoit Cloitre, Mar 02 2005
a(n) = C(2n,n)*C(3n,n) = A104684(2n,n). - Paul Barry, Mar 14 2006
G.f. satisfies: A(x^3) = A( x*(1+3*x+9*x^2)/(1+6*x)^3 )/(1+6*x). - Paul D. Hanna, Oct 29 2010
D-finite with recurrence: n^2*a(n) - 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Dec 04 2012
a(n) = (n+1)^2*(n+2)*A161581(n) for n>2. - Alexander Adamchuk, Dec 27 2013
0 = a(n)^2*(472392*a(n+1)^2 - 83106*a(n+1)*a(n+2) + 3600*a(n+2)^2) + a(n)*a(n+1)*(-8748*a(n+1)^2 + 1953*a(n+1)*a(n+2) - 120*a(n+2)^2) + a(n+1)^2*(36*a(n+1)^2 - 12*a(n+1)*a(n+2) + a(n+2)^2) for all n in Z. - Michael Somos, Oct 22 2014
0 = x*(27*x-1)*y'' + (54*x-1)*y' + 6*y, where y is g.f. - Gheorghe Coserea, Jul 06 2016
From Peter Bala, Jul 15 2016: (Start)
a(n) = 3*binomial(2*n - 1,n)*binomial(3*n - 1,n) = 3*[x^n] 1/(1 - x)^n * [x^n] 1/(1 - x)^(2*n) for n >= 1.
a(n) = binomial(2*n,n)*binomial(3*n,n) = ([x^n](1 + x)^(2*n)) *([x^n](1 + x)^(3*n)) = [x^n](F(x)^(6*n)), where F(x) = 1 + x + 2*x^2 + 14*x^3 + 127*x^4 + 1364*x^5 + 16219*x^6 + ... appears to have integer coefficients. Cf. A002894.
This sequence occurs as the right-hand side of several binomial sums:
Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n,k)^3 = a(n) (Dixon's identity).
Sum_{k = 0..n} binomial(n,k)*binomial(2*n,n - k)*binomial(3*n + k,k) = a(n) (Gould, Vol. 4, 6.86)
Sum_{k = 0..n} (-1)^(n+k)*binomial(n,k)*binomial(2*n + k,n)*binomial(3*n + k,n) = a(n).
Sum_{k = 0..n} binomial(n,k)*binomial(2*n + k,k)*binomial(3*n,n - k) = a(n).
Sum_{k = 0..n} (-1)^(k)*binomial(n,k)*binomial(3*n - k,n)*binomial(4*n - k,n) = a(n).
Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n + k,2*n - k)*binomial(2*k,k)*binomial(4*n - k,2*n) = a(n) (see Gould, Vol.5, 9.23).
Sum_{k = 0..2*n} (-1)^k*binomial(3*n,k)*binomial(3*n - k,n)^3 = a(n) (Sprugnoli, Section 2.9, Table 10, p. 123). (End)
From Bradley Klee, Feb 28 2018: (Start)
a(n) = A005809(n)*A000984(n).
G.f.: F(x) = 1/(2*Pi) Integral_{z=0..2*Pi} 2F1(1/3,2/3; 1/2; 27*x*sin^2(z)) dz.
With G(x) = x*2F1(1/3,2/3; 2; 27*x): F(x) = d/dx G(x). (Cf. A007004) (End)
F(x)*G(1/27-x) + F(1/27-x)*G(x) = 1/(4*Pi*sqrt(3)). - Bradley Klee, Sep 29 2018
Sum_{n>=0} 1/a(n) = A091683. - Amiram Eldar, Nov 15 2020
From Peter Bala, Sep 20 2021: (Start)
a(n) = Sum_{k = n..2*n} binomial(2*n,k)^2 * binomial(k,n). Cf. A001459.
a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for any prime p >= 5 and any positive integers n and k (write a(n) as C(3*n,2*n)*C(2*n,n) and apply Mestrovic, equation 39, p. 12). (End)
a(n) = 6*A060542(n). - R. J. Mathar, Jun 21 2023
Occurs on the right-hand side of the binomial sum identities Sum_{k = -n..n} (-1)^k * (n + x - k) * binomial(2*n, n+k)^3 = (x + n)*a(n) and Sum_{k = -n..n} (-1)^k * (n + x - k)^3 * binomial(2*n, n+k)^3 = x*(x + n)*(x + 2*n)*a(n) (x arbitrary). Compare with Dixon's identity: Sum_{k = -n..n} (-1)^k * binomial(2*n, n+k)^3 = a(n). - Peter Bala, Jul 31 2023
From Peter Bala, Aug 14 2023: (Start)
a(n) = (-1)^n * [x^(2*n)] ( (1 - x)^(4*n) * Legendre_P(2*n, (1 + x)/(1 - x)) ).
Row 1 of A364509. (End)
From Peter Bala, Oct 10 2024: (Start)
The following hold for n >= 1:
a(n) = Sum_{k = 0.. 2*n} (-1)^(n+k) * k/n * binomial(2*n, k)^3 = 3/2 * Sum_{k = 0.. 2*n} (-1)^(n+k) * (k/n)^2 * binomial(2*n, k)^3.
a(n) = 3/2 * Sum_{0..2*n-1} (-1)^(n+k) * k/n * binomial(2*n, k)^2*binomial(2*n-1, k).
a(n) = 3 * Sum_{0..2*n-1} (-1)^(n+k) * k/n * binomial(2*n, k)*binomial(2*n-1, k)^2. (End)
a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n, k) * A108625(2*n, k) (verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package). - Peter Bala, Oct 15 2024

a(14)-a(16) from Eric W. Weisstein

Terms a(17) and beyond from T. D. Noe, Jun 29 2008

A001075 a(0) = 1, a(1) = 2, a(n) = 4*a(n-1) - a(n-2).

Original entry on oeis.org

1, 2, 7, 26, 97, 362, 1351, 5042, 18817, 70226, 262087, 978122, 3650401, 13623482, 50843527, 189750626, 708158977, 2642885282, 9863382151, 36810643322, 137379191137, 512706121226, 1913445293767, 7141075053842, 26650854921601, 99462344632562, 371198523608647

Offset: 0

N. J. A. Sloane

Chebyshev's T(n,x) polynomials evaluated at x=2.
x = 2^n - 1 is prime if and only if x divides a(2^(n-2)).
Any k in the sequence is succeeded by 2*k + sqrt{3*(k^2 - 1)}. - Lekraj Beedassy, Jun 28 2002
For all elements x of the sequence, 12*x^2 - 12 is a square. Lim_{n -> infinity} a(n)/a(n-1) = 2 + sqrt(3) = (4 + sqrt(12))/2 which preserves the kinship with the equation "12*x^2 - 12 is a square" where the initial "12" ends up appearing as a square root. - Gregory V. Richardson, Oct 10 2002
This sequence gives the values of x in solutions of the Diophantine equation x^2 - 3*y^2 = 1; the corresponding values of y are in A001353. The solution ratios a(n)/A001353(n) are obtained as convergents of the continued fraction expansion of sqrt(3): either as successive convergents of [2;-4] or as odd convergents of [1;1,2]. - Lekraj Beedassy, Sep 19 2003 [edited by Jon E. Schoenfield, May 04 2014]
a(n) is half the central value in a list of three consecutive integers, the lengths of the sides of a triangle with integer sides and area. - Eugene McDonnell (eemcd(AT)mac.com), Oct 19 2003
a(3+6*k) - 1 and a(3+6*k) + 1 are consecutive odd powerful numbers. See A076445. - T. D. Noe, May 04 2006
The intermediate convergents to 3^(1/2), beginning with 3/2, 12/7, 45/26, 168/97, comprise a strictly increasing sequence; essentially, numerators=A005320, denominators=A001075. - Clark Kimberling, Aug 27 2008
The upper principal convergents to 3^(1/2), beginning with 2/1, 7/4, 26/15, 97/56, comprise a strictly decreasing sequence; numerators=A001075, denominators=A001353. - Clark Kimberling, Aug 27 2008
a(n+1) is the Hankel transform of A000108(n) + A000984(n) = (n+2)*Catalan(n). - Paul Barry, Aug 11 2009
Also, numbers such that floor(a(n)^2/3) is a square: base 3 analog of A031149, A204502, A204514, A204516, A204518, A204520, A004275, A001541. - M. F. Hasler, Jan 15 2012
Pisano period lengths:  1, 2, 2, 4, 3, 2, 8, 4, 6, 6, 10, 4, 12, 8, 6, 8, 18, 6, 5, 12, ... - R. J. Mathar, Aug 10 2012
Except for the first term, positive values of x (or y) satisfying x^2 - 4*x*y + y^2 + 3 = 0. - Colin Barker, Feb 04 2014
Except for the first term, positive values of x (or y) satisfying x^2 - 14*x*y + y^2 + 48 = 0. - Colin Barker, Feb 10 2014
From Gary W. Adamson, Jul 25 2016: (Start)
A triangle with row sums generating the sequence can be constructed by taking the production matrix M. Take powers of M, extracting the top rows.
M =
  1, 1, 0, 0, 0, 0, ...
  2, 0, 3, 0, 0, 0, ...
  2, 0, 0, 3, 0, 0, ...
  2, 0, 0, 0, 3, 0, ...
  2, 0, 0, 0, 0, 3, ...
  ...
The triangle generated from M is:
   1,
   1,  1,
   3,  1, 3,
  11,  3, 3, 9,
  41, 11, 9, 9, 27,
   ...
The left border is A001835 and row sums are (1, 2, 7, 26, 97, ...). (End)
Even-indexed terms are odd while odd-indexed terms are even. Indeed, a(2*n) = 2*(a(n))^2 - 1 and a(2*n+1) = 2*a(n)*a(n+1) - 2. - Timothy L. Tiffin, Oct 11 2016
For each n, a(0) divides a(n), a(1) divides a(2n+1), a(2) divides a(4*n+2), a(3) divides a(6*n+3), a(4) divides a(8*n+4), a(5) divides a(10n+5), and so on. Thus, a(k) divides a((2*n+1)*k) for each k > 0 and n >= 0. A proof of this can be found in Bhargava-Kedlaya-Ng's first solution to Problem A2 of the 76th Putnam Mathematical Competition. Links to the exam and its solutions can be found below. - Timothy L. Tiffin, Oct 12 2016
From Timothy L. Tiffin, Oct 21 2016: (Start)
If any term a(n) is a prime number, then its index n will be a power of 2. This is a consequence of the results given in the previous two comments. See A277434 for those prime terms.
a(2n) == 1 (mod 6) and a(2*n+1) == 2 (mod 6). Consequently, each odd prime factor of a(n) will be congruent to 1 modulo 6 and, thus, found in A002476.
a(n) == 1 (mod 10) if n == 0 (mod 6), a(n) == 2 (mod 10) if n == {1,-1} (mod 6), a(n) == 7 (mod 10) if n == {2,-2} (mod 6), and a(n) == 6 (mod 10) if n == 3 (mod 6). So, the rightmost digits of a(n) form a repeating cycle of length 6: 1, 2, 7, 6, 7, 2. (End)
a(A298211(n)) = A002350(3*n^2). - A.H.M. Smeets, Jan 25 2018
(2 + sqrt(3))^n = a(n) + A001353(n)*sqrt(3), n >= 0; integers in the quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 16 2018
Yong Hao Ng has shown that for any n, a(n) is coprime with any member of A001834 and with any member of A001835. - René Gy, Feb 26 2018
Positive numbers k such that 3*(k-1)*(k+1) is a square. - Davide Rotondo, Oct 25 2020
a(n)*a(n+1)-1 = a(2*n+1)/2 = A001570(n) divides both a(n)^6+1 and a(n+1)^6+1. In other words, for k = a(2*n+1)/2, (k+1)^6 has divisors congruent to -1 modulo k (cf. A350916). - Max Alekseyev, Jan 23 2022

			2^6 - 1 = 63 does not divide a(2^4) = 708158977, therefore 63 is composite. 2^5 - 1 = 31 divides a(2^3) = 18817, therefore 31 is prime.
G.f. = 1 + 2*x + 7*x^2 + 26*x^3 + 97*x^4 + 362*x^5 + 1351*x^6 + 5042*x^7 + ...

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
Eugene McDonnell, "Heron's Rule and Integer-Area Triangles", Vector 12.3 (January 1996) pp. 133-142.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P.-F. Teilhet, Reply to Query 2094, L'Intermédiaire des Mathématiciens, 10 (1903), 235-238.

Indranil Ghosh, Table of n, a(n) for n = 0..1745 (terms 0..200 from T. D. Noe)
Christian Aebi and Grant Cairns, Lattice Equable Parallelograms, arXiv:2006.07566 [math.NT], 2020.
Christian Aebi and Grant Cairns, Less than Equable Triangles on the Eisenstein lattice, arXiv:2312.10866 [math.CO], 2023.
Krassimir T. Atanassov and Anthony G. Shannon, On intercalated Fibonacci sequences, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 218-223.
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
H. Brocard, Notes élémentaires sur le problème de Pell, Nouvelle Correspondance Mathématique, 4 (1878), 337-343.
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 30, 43, 56.
Chris Caldwell, Primality Proving, Arndt's theorem.
J. B. Cosgrave and K. Dilcher, A role for generalized Fermat numbers, Math. Comp., 2016.
G. Dresden and Y. Li, Periodic Weighted Sums of Binomial Coefficients, arXiv:2210.04322 [math.NT], 2022.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Margherita Maria Ferrari and Norma Zagaglia Salvi, Aperiodic Compositions and Classical Integer Sequences, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.8.
R. K. Guy, Letter to N. J. A. Sloane concerning A001075, A011943, A094347 [Scanned and annotated letter, included with permission]
Kiran S. Kedlaya, The 76th William Lowell Putnam Mathematical Competition, Dec 05 2015.
Kiran S. Kedlaya, Solutions to the 76th William Lowell Putnam Mathematical Competition, Dec 05 2015.
Tanya Khovanova, Recursive Sequences
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.
Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.
Eugene McDonnell, Heron's Rule and Integer-Area Triangles, At Play With J, 2010.
Valcho Milchev and Tsvetelina Karamfilova, Domino tiling in grid - new dependence, arXiv:1707.09741 [math.HO], 2017.
Yong Hao Ng, A partition in three classes of the set of all prime numbers?, Mathematics Stack Exchange.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.
Index entries for sequences related to Chebyshev polynomials.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Bisections are A011943 and A094347.

Cf. A001353, A065918, A071954, A001571, A001834, A003500, A016064, A082840, A026150, A277434.

a001075 n = a001075_list !! n
a001075_list =
   1 : 2 : zipWith (-) (map (4 *) $ tail a001075_list) a001075_list
-- Reinhard Zumkeller, Aug 11 2011

I:=[1, 2]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 19 2017

A001075 := proc(n)
    orthopoly[T](n,2) ;
end proc:
seq(A001075(n),n=0..30) ; # R. J. Mathar, Apr 14 2018

Table[ Ceiling[(1/2)*(2 + Sqrt[3])^n], {n, 0, 24}]
CoefficientList[Series[(1-2*x) / (1-4*x+x^2), {x, 0, 24}], x] (* Jean-François Alcover, Dec 21 2011, after Simon Plouffe *)
LinearRecurrence[{4,-1},{1,2},30] (* Harvey P. Dale, Aug 22 2015 *)
Round@Table[LucasL[2n, Sqrt[2]]/2, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
ChebyshevT[Range[0, 20], 2] (* Eric W. Weisstein, May 26 2017 *)
a[ n_] := LucasL[2*n, x]/2 /. x->Sqrt[2]; (* Michael Somos, Sep 05 2022 *)

{a(n) = subst(poltchebi(abs(n)), x, 2)};

{a(n) = real((2 + quadgen(12))^abs(n))};

{a(n) = polsym(1 - 4*x + x^2, abs(n))[1 + abs(n)]/2};

a(n)=polchebyshev(n,1,2) \\ Charles R Greathouse IV, Nov 07 2016

my(x='x+O('x^30)); Vec((1-2*x)/(1-4*x+x^2)) \\ G. C. Greubel, Dec 19 2017

[lucas_number2(n,4,1)/2 for n in range(0, 25)] # Zerinvary Lajos, May 14 2009

def a(n):
    Q = QuadraticField(3, 't')
    u = Q.units()[0]
    return (u^n).lift().coeffs()[0]  # Ralf Stephan, Jun 19 2014

G.f.: (1 - 2*x)/(1 - 4*x + x^2). - Simon Plouffe in his 1992 dissertation
E.g.f.: exp(2*x)*cosh(sqrt(3)*x).
a(n) = 4*a(n-1) - a(n-2) = a(-n).
a(n) = (S(n, 4) - S(n-2, 4))/2 = T(n, 2), with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. U, resp. T, are Chebyshev's polynomials of the second, resp. first, kind. S(n-1, 4) = A001353(n), n >= 0. See A049310 and A053120.
a(n) = A001353(n+2) - 2*A001353(n+1).
a(n) = sqrt(1 + 3*A001353(n)) (cf. Richardson comment, Oct 10 2002).
a(n) = 2^(-n)*Sum_{k>=0} binomial(2*n, 2*k)*3^k = 2^(-n)*Sum_{k>=0} A086645(n, k)*3^k. - Philippe Deléham, Mar 01 2004
a(n) = ((2 + sqrt(3))^n + (2 - sqrt(3))^n)/2; a(n) = ceiling((1/2)*(2 + sqrt(3))^(n)).
a(n) = cosh(n * log(2 + sqrt(3))).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*2^(n-2*k)*3^k. - Paul Barry, May 08 2003
a(n+2) = 2*a(n+1) + 3*Sum_{k>=0} a(n-k)*2^k. - Philippe Deléham, Mar 03 2004
a(n) = 2*a(n-1) + 3*A001353(n-1). - Lekraj Beedassy, Jul 21 2006
a(n) = left term of M^n * [1,0] where M = the 2 X 2 matrix [2,3; 1,2]. Right term = A001353(n). Example: a(4) = 97 since M^4 * [1,0] = [A001075(4), A001353(4)] = [97, 56]. - Gary W. Adamson, Dec 27 2006
Binomial transform of A026150: (1, 1, 4, 10, 28, 76, ...). - Gary W. Adamson, Nov 23 2007
First differences of A001571. - N. J. A. Sloane, Nov 03 2009
Sequence satisfies -3 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 4*u*v. - Michael Somos, Sep 19 2008
a(n) = Sum_{k=0..n} A201730(n,k)*2^k. - Philippe Deléham, Dec 06 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(3*k - 4)/(x*(3*k - 1) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013
a(n) = Sum_{k=0..n} A238731(n,k). - Philippe Deléham, Mar 05 2014
a(n) = (-1)^n*(A125905(n) + 2*A125905(n-1)), n > 0. - Franck Maminirina Ramaharo, Nov 11 2018
a(n) = (tan(Pi/12)^n + tan(5*Pi/12)^n)/2. - Greg Dresden, Oct 01 2020
From Peter Bala, Aug 17 2022: (Start)
a(n) = (1/2)^n * [x^n] ( 4*x + sqrt(1 + 12*x^2) )^n.
The g.f. A(x) satisfies A(2*x) = 1 + x*B'(x)/B(x), where B(x) = 1/sqrt(1 - 8*x + 4*x^2) is the g.f. of A069835.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p >= 3 and positive integers n and k.
Sum_{n >= 1} 1/(a(n) - (3/2)/a(n)) = 1.
Sum_{n >= 1} (-1)^(n+1)/(a(n) + (1/2)/a(n)) = 1/3.
Sum_{n >= 1} 1/(a(n)^2 - 3/2) = 1 - 1/sqrt(3). (End)
a(n) = binomial(2*n, n) + 2*Sum_{k > 0} binomial(2*n, n+2*k)*cos(k*Pi/3). - Greg Dresden, Oct 11 2022
2*a(n) + 2^n = 3*Sum_{k=-n..n} (-1)^k*binomial(2*n, n+6*k). - Greg Dresden, Feb 07 2023

More terms from James Sellers, Jul 10 2000

Chebyshev comments from Wolfdieter Lang, Oct 31 2002

cp's OEIS Frontend

A005259 Apery (Apéry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2.

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A005809 a(n) = binomial(3n,n).

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A039599 Triangle formed from even-numbered columns of triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

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A002894 a(n) = binomial(2n, n)^2.

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A106529 Numbers having k prime factors (counted with multiplicity), the largest of which is the k-th prime.

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A027383 a(2n) = 32^n - 2; a(2*n+1) = 2^(n+2) - 2.