A143007 -id:A143007 - cp's OEIS frontend

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

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Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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

A002893 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2*k,k).

Original entry on oeis.org

1, 3, 15, 93, 639, 4653, 35169, 272835, 2157759, 17319837, 140668065, 1153462995, 9533639025, 79326566595, 663835030335, 5582724468093, 47152425626559, 399769750195965, 3400775573443089, 29016970072920387, 248256043372999089

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
a(n) is the 2n-th moment of the distance from the origin of a 3-step random walk in the plane. - Peter M. W. Gill (peter.gill(AT)nott.ac.uk), Feb 27 2004
a(n) is the number of Abelian squares of length 2n over a 3-letter alphabet. - Jeffrey Shallit, Aug 17 2010
Consider 2D simple random walk on honeycomb lattice. a(n) gives number of paths of length 2n ending at origin. - Sergey Perepechko, Feb 16 2011
Row sums of A318397 the square of A008459. - Peter Bala, Mar 05 2013
Conjecture: For each n=1,2,3,... the polynomial g_n(x) = Sum_{k=0..n} binomial(n,k)^2*binomial(2k,k)*x^k is irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 21 2013
This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
a(n) is the sum of the squares of the coefficients of (x + y + z)^n. - Michael Somos, Aug 25 2018
a(n) is the constant term in the expansion of (1 + (1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 28 2019

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 639*x^4 + 4653*x^5 + 35169*x^6 + ...
G.f.: A(x) = 1/(1-3*x) + 6*x^2*(1-x)/(1-3*x)^4 + 90*x^4*(1-x)^2/(1-3*x)^7 + 1680*x^6*(1-x)^3/(1-3*x)^10 + 34650*x^8*(1-x)^4/(1-3*x)^13 + ... - _Paul D. Hanna_, Feb 26 2012

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..1051 (terms 0..100 from T. D. Noe)
B. Adamczewski, J. P. Bell, and E. Delaygue, Algebraic independence of G-functions and congruences à la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
Rostislav Akhmechet, Mikhail Khovanov, and Melissa Zhang, Annular SL(2) and SL(3) web algebras, arXiv:2508.05605 [math.GT], 2025. See p. 48.
Gert Almkvist, The art of finding Calabi-Yau differential equations, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008.
P. Barrucand, A combinatorial identity, Problem 75-4, SIAM Rev., 17 (1975), 168. Solution by D. R. Breach, D. McCarthy, D. Monk and P. E. O'Neil, SIAM Rev. 18 (1976), 303.
P. Barrucand, Problem 75-4, A Combinatorial Identity, SIAM Rev., 17 (1975), 168. [Annotated scanned copy of statement of problem]
Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982.
Frits Beukers and Jan Stienstra, On the Picard-Fuchs Equation and the Formal Brauer Group of Certain Elliptic K3-Surfaces, Mathematische Annalen (1985), Vol. 271, pp. 269-304 (see Part III).
Artur Bille, Victor Buchstaber, Simon Coste, Satoshi Kuriki, and Evgeny Spodarev, Random eigenvalues of graphenes and the triangulation of plane, arXiv:2306.01462 [math.SP], 2023.
Jonathan M. Borwein, A short walk can be beautiful, 2015.
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals.
Jonathan M. Borwein, Armin Straub and Christophe Vignat, Densities of short uniform random walks, Part II: Higher dimensions, Preprint, 2015
Jonathan M. Borwein, Armin Straub and James Wan, Three-Step and Four-Step Random Walk Integrals, Exper. Math., 22 (2013), 1-14.
Charles Burnette and Chung Wong, Abelian Squares and Their Progenies, arXiv:1609.05580 [math.CO], 2016.
David Callan, A combinatorial interpretation for an identity of Barrucand, JIS 11 (2008) 08.3.4.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.
M. Coster, Email, Nov 1990
Eric Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
Jeffrey S. Geronimo, Hugo J. Woerdeman, and Chung Y. Wong, The autoregressive filter problem for multivariable degree one symmetric polynomials, arXiv:2101.00525 [math.CA], 2021.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See C p. 2.
Victor J. W. Guo, Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers, arXiv preprint arXiv:1201.0617 [math.NT], 2012.
Victor J. W. Guo, Guo-Shuai Mao and Hao Pan, Proof of a conjecture involving Sun polynomials, arXiv preprint arXiv:1511.04005 [math.NT], 2015.
E. Hallouin and M. Perret, A Graph Aided Strategy to Produce Good Recursive Towers over Finite Fields, arXiv preprint arXiv:1503.06591 [math.NT], 2015.
J. A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, Journal of Statistical Computation and Simulation, 51 (1995), 291-313.
S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.
Pakawut Jiradilok and Elchanan Mossel, Gaussian Broadcast on Grids, arXiv:2402.11990 [cs.IT], 2024. See p. 27.
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.
Tanya Khovanova and Konstantin Knop, Coins of three different weights, arXiv:1409.0250 [math.HO], 2014.
Murray S. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 148-149.
Bradley Klee, Checking Weierstrass data, 2023.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
Mathematics Stack Exchange, sum involving the product of binomial coefficients, Nov 10 2016.
L. B. Richmond and Jeffrey Shallit, Counting abelian squares, Electronic J. Combinatorics 16 (1), #R72, June 2009. [From _Jeffrey Shallit_, Aug 17 2010]
Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012. - From _N. J. A. Sloane_, Dec 16 2012
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, Connections between p = x^2+3y^2 and Franel numbers, J. Number Theory 133(2013), 2919-2928.
Zhi-Wei Sun, Congruences involving g_n(x) = Sum_{k=0..n} binomial(n,k)^2*binomial(2k,k)*x^k, Ramanujan J., in press. Doi: 10.1007/s11139-015-9727-3.
Brani Vidakovic, All roads lead to Rome--even in the honeycomb world, Amer. Statist., 48 (1994) no. 3, 234-236.
Yi Wang and Bao-Xuan Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, arXiv preprint arXiv:1303.5595 [math.CO], 2013.
D. Zagier, Integral solutions of Apery-like recurrence equations. See line C in sporadic solutions table of page 5.

Cf. A000172, A002895, A000984, A006480, A087457, A274600, A318397, A244973.
Cf. A169714, A169715.
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.

[&+[Binomial(n, k)^2 * Binomial(2*k, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 26 2018

series(1/GaussAGM(sqrt((1-3*x)*(1+x)^3), sqrt((1+3*x)*(1-x)^3)), x=0, 42) # Gheorghe Coserea, Aug 17 2016
A002893 := n -> hypergeom([1/2, -n, -n], [1, 1], 4):
seq(simplify(A002893(n)), n=0..20); # Peter Luschny, May 23 2017

Table[Sum[Binomial[n,k]^2 Binomial[2k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Aug 19 2011 *)
a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {1/2, -n, -n}, {1, 1}, 4]]; (* Michael Somos, Oct 16 2013 *)
a[n_] := SeriesCoefficient[BesselI[0, 2*Sqrt[x]]^3, {x, 0, n}]*n!^2; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 30 2013 *)
a[ n_] := If[ n < 0, 0, Block[ {x, y, z},  Expand[(x + y + z)^n] /. {t_Integer -> t^2, x -> 1, y -> 1, z -> 1}]]; (* Michael Somos, Aug 25 2018 *)

{a(n) = if( n<0, 0, n!^2 * polcoeff( besseli(0, 2*x + O(x^(2*n+1)))^3, 2*n))};

{a(n) = sum(k=0, n, binomial(n, k)^2 * binomial(2*k, k))}; /* Michael Somos, Jul 25 2007 */

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3 * x^(2*m)*(1-x)^m / (1-3*x+x*O(x^n))^(3*m+1)),n)} \\ Paul D. Hanna, Feb 26 2012

N = 42; x='x + O('x^N); v = Vec(1/agm(sqrt((1-3*x)*(1+x)^3), sqrt((1+3*x)*(1-x)^3))); vector((#v+1)\2, k, v[2*k-1])  \\ Gheorghe Coserea, Aug 17 2016

def A002893(n): return simplify(hypergeometric([1/2,-n,-n], [1,1], 4))
[A002893(n) for n in range(31)] # G. C. Greubel, Jan 21 2023

a(n) = Sum_{m=0..n} binomial(n, m) * A000172(m). [Barrucand]
D-finite with recurrence: (n+1)^2 a(n+1) = (10*n^2+10*n+3) * a(n) - 9*n^2 * a(n-1). - Matthijs Coster, Apr 28 2004
Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0, 2*sqrt(x))^3. - Vladeta Jovovic, Mar 11 2003
a(n) = Sum_{p+q+r=n} (n!/(p!*q!*r!))^2 with p, q, r >= 0. - Michael Somos, Jul 25 2007
a(n) = 3*A087457(n) for n>0. - Philippe Deléham, Sep 14 2008
a(n) = hypergeom([1/2, -n, -n], [1, 1], 4). - Mark van Hoeij, Jun 02 2010
G.f.: 2*sqrt(2)/Pi/sqrt(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z))) *  EllipticK(8*z^(3/2)/(1-6*z-3*z^2+sqrt((1-z)^3*(1-9*z)))). - Sergey Perepechko, Feb 16 2011
G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1-x)^n / (1-3*x)^(3*n+1). - Paul D. Hanna, Feb 26 2012
Asymptotic: a(n) ~ 3^(2*n+3/2)/(4*Pi*n). - Vaclav Kotesovec, Sep 11 2012
G.f.: 1/(1-3*x)*(1-6*x^2*(1-x)/(Q(0)+6*x^2*(1-x))), where Q(k) = (54*x^3 - 54*x^2 + 9*x -1)*k^2 + (81*x^3 - 81*x^2 + 18*x -2)*k + 33*x^3 - 33*x^2 +9*x - 1 - 3*x^2*(1-x)*(1-3*x)^3*(k+1)^2*(3*k+4)*(3*k+5)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
G.f.: G(0)/(2*(1-9*x)^(2/3)), where G(k) = 1 + 1/(1 - 3*(3*k+1)^2*x*(1-x)^2/(3*(3*k+1)^2*x*(1-x)^2 - (k+1)^2*(1-9*x)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 31 2013
a(n) = [x^(2n)] 1/agm(sqrt((1-3*x)*(1+x)^3), sqrt((1+3*x)*(1-x)^3)). - Gheorghe Coserea, Aug 17 2016
0 = +a(n)*(+a(n+1)*(+729*a(n+2) -1539*a(n+3) +243*a(n+4)) +a(n+2)*(-567*a(n+2) +1665*a(n+3) -297*a(n+4)) +a(n+3)*(-117*a(n+3) +27*a(n+4))) +a(n+1)*(+a(n+1)*(-324*a(n+2) +720*a(n+3) -117*a(n+4)) +a(n+2)*(+315*a(n+2) -1000*a(n+3) +185*a(n+4)) +a(n+3)*(+80*a(n+3) -19*a(n+4))) +a(n+2)*(+a(n+2)*(-9*a(n+2) +35*a(n+3) -7*a(n+4)) +a(n+3)*(-4*a(n+3) +a(n+4))) for all n in Z. - Michael Somos, Oct 30 2017
G.f. y=A(x) satisfies: 0 = x*(x - 1)*(9*x - 1)*y'' + (27*x^2 - 20*x + 1)*y' + 3*(3*x - 1)*y. - Gheorghe Coserea, Jul 01 2018
Sum_{k>=0} binomial(2*k,k) * a(k) / 6^(2*k) = A086231 = (sqrt(3)-1) * (Gamma(1/24) * Gamma(11/24))^2 / (32*Pi^3). - Vaclav Kotesovec, Apr 23 2023
From Bradley Klee, Jun 05 2023: (Start)
The g.f. T(x) obeys a period-annihilating ODE:
0=3*(-1 + 3*x)*T(x) + (1 - 20*x + 27*x^2)*T'(x) + x*(-1 + x)*(-1 + 9*x)*T''(x).
The periods ODE can be derived from the following Weierstrass data:
g2 = (3/64)*(1 + 3*x)*(1 - 15*x + 75*x^2 + 3*x^3);
g3 = -(1/512)*(-1 + 6*x + 3*x^2)*(1 - 12*x + 30*x^2 - 540*x^3 + 9*x^4);
which determine an elliptic surface with four singular fibers. (End)
a(n) = Sum_{k = 0..n} binomial(n, k)^2 * binomial(3*k, 2*n) (Almkvist, p. 16). - Peter Bala, May 22 2025

A002895 Domb numbers: number of 2n-step polygons on diamond lattice.

Original entry on oeis.org

1, 4, 28, 256, 2716, 31504, 387136, 4951552, 65218204, 878536624, 12046924528, 167595457792, 2359613230144, 33557651538688, 481365424895488, 6956365106016256, 101181938814289564, 1480129751586116848, 21761706991570726096, 321401321741959062016

Offset: 0

N. J. A. Sloane

a(n) is the (2n)th moment of the distance from the origin of a 4-step random walk in the plane. - Peter M.W. Gill (peter.gill(AT)nott.ac.uk), Mar 03 2004
Row sums of the cube of A008459. - Peter Bala, Mar 05 2013
Conjecture: Let D(n) be the (n+1) X (n+1) Hankel-type determinant with (i,j)-entry equal to a(i+j) for all i,j = 0..n. Then the number D(n)/12^n is always a positive odd integer. - Zhi-Wei Sun, Aug 14 2013
It appears that the expansions exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 4*x + 22*x^2 + 152*x^3 + 1241*x^4 + ... and exp( Sum_{n >= 1} 1/4*a(n)*x^n/n ) = 1 + x + 4*x^2 + 25*x^3 + 199*x^4 + ... have integer coefficients. See A267219. - Peter Bala, Jan 12 2016
This is one of the Apéry-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
Named after the British-Israeli theoretical physicist Cyril Domb (1920-2012). - Amiram Eldar, Mar 20 2021

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).

Indranil Ghosh, Table of n, a(n) for n = 0..832 (terms 0..100 from T. D. Noe)
B. Adamczewski, Jason P. Bell and E. Delaygue, Algebraic independence of G-functions and congruences "a la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments and applications, Journal of Physics A: Mathematical and Theoretical, Vol. 41, No. 20 (2008), 205203; arXiv preprint, arXiv:0801.0891 [hep-th], 2008.
Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See p. 20.
Jonathan M. Borwein, A short walk can be beautiful, Journal of Humanistic Mathematics, Vol. 6, No. 1 (2016), pp. 86-109; preprint, 2015.
Jonathan M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
Jonathan M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals, Theoretical Computer Science, Vol. 479 (2013), pp. 4-21.
Jonathan M. Borwein, Armin Straub and Christophe Vignat, Densities of short uniform random walks, Part II: Higher dimensions, Preprint, 2015.
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
H. Huat Chan, Song Heng Chan and Zhiguo Liu, Domb's numbers and Ramanujan-Sato type series for 1/pi, Adv. Math., Vol. 186, No. 2 (2004), pp. 396-410.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023. See Table 2 p. 7.
Shaun Cooper, James G. Wan and Wadim Zudilin, Holonomic Alchemy and Series for 1/pi, in: G. Andrews and F. Garvan (eds.) Analytic Number Theory, Modular Forms and q-Hypergeometric Series, ALLADI60 2016, Springer Proceedings in Mathematics & Statistics, Vol 221. Springer, Cham, 2016; arXiv preprint, arXiv:1512.04608 [math.NT], 2015.
Eric Delaygue, Arithmetic properties of Apéry-like numbers, Compositio Mathematica, Vol. 154, No. 2 (2018), pp. 249-274; arXiv preprint, arXiv:1310.4131 [math.NT], 2013-2015.
Cyril Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., Vol. 9 (1960), pp. 149-361.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See alpha p. 3.
John A. Hendrickson, Jr., On the enumeration of rectangular (0,1)-matrices, Journal of Statistical Computation and Simulation, Vol. 51 (1995), pp. 291-313.
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Pakawut Jiradilok and Elchanan Mossel, Gaussian Broadcast on Grids, arXiv:2402.11990 [cs.IT], 2024. See p. 27.
Ji-Cai Liu, Supercongruences for sums involving Domb numbers, arXiv:2008.02647 [math.NT], 2020.
Rui-Li Liu and Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
Yen Lee Loh, A general method for calculating lattice green functions on the branch cut, Journal of Physics A: Mathematical and Theoretical, Vol. 50, No. 40 (2017), 405203; arXiv preprint, arXiv:1706.03083 [math-ph], 2017.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
Guo-Shuai Mao and Yan Liu, Proof of some conjectural congruences involving Domb numbers, arXiv:2112.00511 [math.NT], 2021.
Guo-Shuai Mao and Michael J. Schlosser, Supercongruences involving Domb numbers and binary quadratic forms, arXiv:2112.12732 [math.NT], 2021.
Robert Osburn and Brundaban Sahu, A supercongruence for generalized Domb numbers, Functiones et Approximatio Commentarii Mathematici, Vol. 48, No. 1 (2013), pp. 29-36; preprint.
L. B. Richmond and Jeffrey Shallit, Counting Abelian Squares, The Electronic Journal of Combinatorics, Vol. 16, No. 1 (2009), Article R72; arXiv preprint, arXiv:0807.5028 [math.CO], 2008.
Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012.
Zhi-Hong Sun, Super congruences concerning binomial coefficients and Apéry-like numbers, arXiv:2002.12072 [math.NT], 2020.
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.
Zhi-Wei Sun, Conjectures involving arithmetical sequences, in: S. Kanemitsu, H. Li and J. Liu (eds.), Number Theory: Arithmetic in Shangri-La, Proc. the 6th China-Japan Sem. Number Theory (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244-258; alternative link.
H. A. Verrill, Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations, arXiv:math/0407327 [math.CO], 2004.
Chen Wang, Supercongruences and hypergeometric transformations, arXiv:2003.09888 [math.NT], 2020.
Yi Wang and BaoXuan Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, Science China Mathematics, Vol. 57, No. 11 (2014), pp. 2429-2435; arXiv preprint, arXiv:1303.5595 [math.CO], 2013.
Bao-Xuan Zhu, Higher order log-monotonicity of combinatorial sequences, arXiv preprint, arXiv:1309.6025 [math.CO], 2013.

Cf. A002893, A008459, A169714, A169715, A228289, A267219.
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.

A002895 := n -> add(binomial(n,k)^2*binomial(2*n-2*k,n-k)*binomial(2*k,k), k=0..n): seq(A002895(n), n=0..25); # Wesley Ivan Hurt, Dec 20 2015
A002895 := n -> binomial(2*n,n)*hypergeom([1/2, -n, -n, -n],[1, 1, 1/2 - n], 1):
seq(simplify(A002895(n)), n=0..19); # Peter Luschny, May 23 2017

Table[Sum[Binomial[n,k]^2 Binomial[2n-2k,n-k]Binomial[2k,k],{k,0,n}], {n,0,30}] (* Harvey P. Dale, Aug 15 2011 *)
a[n_] = Binomial[2*n, n]*HypergeometricPFQ[{1/2, -n, -n, -n}, {1, 1, 1/2-n}, 1]; (* or *) a[n_] := SeriesCoefficient[BesselI[0, 2*Sqrt[x]]^4, {x, 0, n}]*n!^2; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Dec 30 2013, after Vladeta Jovovic *)
max = 19; Total /@ MatrixPower[Table[Binomial[n, k]^2, {n, 0, max}, {k, 0, max}], 3] (* Jean-François Alcover, Mar 24 2015, after Peter Bala *)

C=binomial;
a(n) = sum(k=0,n, C(n,k)^2 * C(2*n-2*k,n-k) * C(2*k,k) );
/* Joerg Arndt, Apr 19 2013 */

from math import comb
def A002895(n): return (sum(comb(n,k)**2*comb(n-k<<1,n-k)*comb(m:=k<<1,k) for k in range(n+1>>1))<<1) + (0 if n&1 else comb(n,n>>1)**4) # Chai Wah Wu, Jun 17 2025

a(n) = Sum_{k=0..n} binomial(n, k)^2 * binomial(2n-2k, n-k) * binomial(2k, k).
D-finite with recurrence: n^3*a(n) = 2*(2*n-1)*(5*n^2-5*n+2)*a(n-1) - 64*(n-1)^3*a(n-2). - Vladeta Jovovic, Jul 16 2004
Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0, 2*sqrt(x))^4. - Vladeta Jovovic, Aug 01 2006
G.f.: hypergeom([1/6, 1/3],[1],108*x^2/(1-4*x)^3)^2/(1-4*x). - Mark van Hoeij, Oct 29 2011
From Zhi-Wei Sun, Mar 20 2013: (Start)
Via the Zeilberger algorithm, Zhi-Wei Sun proved that:
(1) 4^n*a(n) = Sum_{k = 0..n} (binomial(2k,k)*binomial(2(n-k),n-k))^3/ binomial(n,k)^2,
(2) a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*binomial(2k,n)*binomial(2k,k)* binomial(2(n-k),n-k). (End)
a(n) ~ 2^(4*n+1)/((Pi*n)^(3/2)). - Vaclav Kotesovec, Aug 20 2013
G.f. y=A(x) satisfies: 0 = x^2*(4*x - 1)*(16*x - 1)*y''' + 3*x*(128*x^2 - 30*x + 1)*y'' + (448*x^2 - 68*x + 1)*y' + 4*(16*x - 1)*y. - Gheorghe Coserea, Jun 26 2018
a(n) = Sum_{p+q+r+s=n} (n!/(p!*q!*r!*s!))^2 with p,q,r,s >= 0. See Verrill, p. 5. - Peter Bala, Jan 06 2020
From Peter Bala, Jul 25 2024: (Start)
a(n) = 2*Sum_{k = 1..n} (k/n)*binomial(n, k)^2*binomial(2*n-2*k, n-k)* binomial(2*k, k) for n >= 1.
a(n-1) = (1/2)*Sum_{k = 1..n} (k/n)^3*binomial(n, k)^2*binomial(2*n-2*k, n-k)* binomial(2*k, k) for n >= 1. Cf. A081085. (End)

More terms from Vladeta Jovovic, Mar 11 2003

A005260 a(n) = Sum_{k = 0..n} binomial(n,k)^4.

Original entry on oeis.org

1, 2, 18, 164, 1810, 21252, 263844, 3395016, 44916498, 607041380, 8345319268, 116335834056, 1640651321764, 23365271704712, 335556407724360, 4854133484555664, 70666388112940818, 1034529673001901732, 15220552520052960516, 224929755893153896200, 3337324864503769353060

Offset: 0

N. J. A. Sloane

This sequence is s_10 in Cooper's paper. - Jason Kimberley, Nov 25 2012
Diagonal of the rational function R(x,y,z,w) = 1/(1 - (w*x*y + w*x*z + w*y*z + x*y*z + w*x + y*z)). - Gheorghe Coserea, Jul 13 2016
This is one of the Apéry-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
Every prime eventually divides some term of this sequence. - Amita Malik, Aug 20 2017
Two walkers, A and B, stand on the South-West and North-East corners of an n X n grid, respectively. A walks by either North or East steps while B walks by either South or West steps. Sequence values a(n) < binomial(2*n,n)^2 count the simultaneous walks where A and B meet after exactly n steps and change places after 2*n steps. - Bradley Klee, Apr 01 2019
a(n) is the constant term in the expansion of ((1 + x) * (1 + y) * (1 + z) + (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n. - Seiichi Manyama, Oct 27 2019

			G.f. = 1 + 2*x + 18*x^2 + 164*x^3 + 1810*x^4 + 21252*x^5 + 263844*x^6 + ...

H. W. Gould, Combinatorial Identities, Morgantown, 1972, (X.14), p. 79.
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..834 (terms 0..250 from Jason Kimberley)
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.
Hacene Belbachir and Yassine Otmani, A Strehl Version of Fourth Franel Sequence, arXiv:2012.02563 [math.CO], 2020.
F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201-210.
W. Y. C. Chen, Q.-H. Hou, and Y-P. Mu, A telescoping method for double summations, J. Comp. Appl. Math. 196 (2006) 553-566, eq (5.5).
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
M. Coster, Email, Nov 1990
E. Delaygue, Arithmetic properties of Apéry-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See s10 p. 3.
Darij Grinberg, Introduction to Modern Algebra (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
Robert Osburn, Armin Straub, and Wadim Zudilin, A modular supercongruence for 6F5: an Apéry-like story, arXiv:1701.04098 [math.NT], 2017.
M. A. Perlstadt, Some Recurrences for Sums of Powers of Binomial Coefficients, Journal of Number Theory 27 (1987), pp. 304-309.
V. Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
Zhi-Wei Sun, Open conjectures on congruences, Nanjing Univ. J. Math. Biquarterly 36(2019), no.1, 1-99. (Cf. Conjectures 49-51.)
Eric Weisstein's World of Mathematics, Binomial Sums
Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, preprint of Combinatorics, Probability and Computing, 24(1), 2015, 354-372.
Jin Yuan, Zhi-Juan Lu, and Asmus L. Schmidt, On recurrences for sums of powers of binomial coefficients, J. Numb. Theory 128 (2008) 2784-2794.

Column k=4 of A309010.
Cf. A000172, A096192, A328725.
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.)
Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
Row sums of A202750.

A005260 := proc(n)
        add( (binomial(n,k))^4,k=0..n) ;
end proc:
seq(A005260(n),n=0..10) ; # R. J. Mathar, Nov 19 2012

Table[Sum[Binomial[n, k]^4, {k, 0, n}], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 09 2014 *)

{a(n) = sum(k=0, n, binomial(n, k)^4)};

def A005260(n):
    m, g = 1, 0
    for k in range(n+1):
        g += m
        m = m*(n-k)**4//(k+1)**4
    return g # Chai Wah Wu, Oct 04 2022

a(n) ~ 2^(1/2)*Pi^(-3/2)*n^(-3/2)*2^(4*n). - Joe Keane (jgk(AT)jgk.org), Jun 21 2002
D-finite with recurrence: n^3*a(n) = 2*(2*n - 1)*(3*n^2 - 3*n + 1)*a(n-1) + (4*n - 3)*(4*n - 4)*(4*n - 5)*a(n-2). [Yuan]
G.f.: 5*hypergeom([1/8, 3/8],[1], (4/5)*((1-16*x)^(1/2)+(1+4*x)^(1/2))*(-(1-16*x)^(1/2)+(1+4*x)^(1/2))^5/(2*(1-16*x)^(1/2)+3*(1+4*x)^(1/2))^4)^2/(2*(1-16*x)^(1/2)+3*(1+4*x)^(1/2)). - Mark van Hoeij, Oct 29 2011
1/Pi = sqrt(15)/18 * Sum_{n >= 0} a(n)*(4*n + 1)/36^n (Cooper, equation (5)) = sqrt(15)/18 * Sum_{n >= 0} a(n)*A016813(n)/A009980(n). - Jason Kimberley, Nov 26 2012
0 = (-x^2 + 12*x^3 + 64*x^4)*y''' + (-3*x + 54*x^2 + 384*x^3)*y'' + (-1 + 40*x + 444*x^2)*y' + (2 + 60*x)*y, where y is g.f. - Gheorghe Coserea, Jul 13 2016
For r a nonnegative integer, Sum_{k = r..n} C(k,r)^4*C(n,k)^4 = C(n,r)^4*a(n-r), where we take a(n) = 0 for n < 0. - Peter Bala, Jul 27 2016
a(n) = hypergeom([-n, -n, -n, -n], [1, 1, 1], 1). - Peter Luschny, Jul 27 2016
Sum_{n>=0} a(n) * x^n / (n!)^4 = (Sum_{n>=0} x^n / (n!)^4)^2. - Ilya Gutkovskiy, Jul 17 2020
a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k)*C(2k,k)*C(2n-2k,n-k)*(-1)^(n-k). This can be proved via the Zeilberger algorithm. - Zhi-Wei Sun, Aug 23 2020
a(n) = (-1)^n*binomial(2*n, n)*hypergeom([1/2, -n, -n, n + 1], [1, 1, 1/2 - n], 1). - Peter Luschny, Aug 24 2020
a(n) = Sum_{k=0..n} binomial(n,k)^2*binomial(2*k,n)*binomial(2*n-k,n) [Theorem 1 in Belbachir and Otmani]. - Michel Marcus, Dec 06 2020
a(n) = [x^n] (1 - x)^(2*n) P(n,(1 + x)/(1 - x))^2, where P(n,x) denotes the n-th Legendre polynomial. See Gould, p. 66. This formula is equivalent to the binomial sum identity of Zhi-Wei Sun given above. - Peter Bala, Mar 24 2022
From Peter Bala, Oct 31 2024: (Start)
For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^3 * binomial(n-1, k).
For n >= 1, a(n) = 2 * hypergeom([-n, -n, -n, -n + 1], [1, 1, 1], 1). (End)
G.f.: Sum_{k>=0} Sum_{l=0..p*k} Sum_{m=0..l} (-1)^m*binomial(p*k+1,m)*binomial(l+k-m,k)^p*x^(l+k)/(1-x)^(p*k+1), where p = 4. - Miles Wilson, Apr 12 2025

Edited by Michael Somos, Aug 09 2002

Minor edits by Vaclav Kotesovec, Aug 28 2014

A063007 T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 12, 30, 20, 1, 20, 90, 140, 70, 1, 30, 210, 560, 630, 252, 1, 42, 420, 1680, 3150, 2772, 924, 1, 56, 756, 4200, 11550, 16632, 12012, 3432, 1, 72, 1260, 9240, 34650, 72072, 84084, 51480, 12870, 1, 90, 1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620

Offset: 0

Henry Bottomley, Jul 02 2001

T(n,k) is the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's Cluster algebra of finite type B_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of triangle A008459 (squares of binomial coefficients). For example, x^2+6*x+6 = y^2+4*y+1. - Paul Boddington, Mar 07 2003
T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k N=(0,1) steps. E.g. T(2,0)=1 because we have DD; T(2,1) = 6 because we have NED, NDE, EDN, END, DEN and DNE; T(2,2)=6 because we have NNEE, NENE, NEEN, EENN, ENEN and ENNE. - Emeric Deutsch, Apr 20 2004
Another version of [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] = 1; 1, 0; 1, 2, 0; 1, 6, 6, 0; 1, 12, 30, 20, 0; ..., where DELTA is the operator defined in A084938. - Philippe Deléham Apr 15 2005
Terms in row n are the coefficients of the Legendre polynomial P(n,2x+1) with increasing powers of x.
From Peter Bala, Oct 28 2008: (Start)
Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type B_n (a cyclohedron) [Fomin & Reading, p.60]. See A008459 for the corresponding h-vectors for associahedra of type B_n and A001263 and A033282 respectively for the h-vectors and f-vectors for associahedra of type A_n.
An alternative description of this triangle in terms of f-vectors is as follows. Let A_n be the root lattice generated as a monoid by {e_i - e_j: 0 <= i,j <= n+1}. Let P(A_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the f-vectors of a unimodular triangulation of P(A_n) [Ardila et al.]. A008459 is the corresponding array of h-vectors for these type A_n polytopes. See A127674 (without the signs) for the array of f-vectors for type C_n polytopes and A108556 for the array of f-vectors associated with type D_n polytopes.
The S-transform on the ring of polynomials is the linear transformation of polynomials that is defined on the basis monomials x^k by S(x^k) = binomial(x,k) = x(x-1)...(x-k+1)/k!. Let P_n(x) denote the S-transform of the n-th row polynomial of this array. In the notation of [Hetyei] these are the Stirling polynomials of the type B associahedra. The first few values are P_1(x) = 2*x + 1, P_2(x) = 3*x^2 + 3*x + 1 and P_3(x) = (10*x^3 + 15*x^2 + 11*x + 3)/3. These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane, that is, the polynomials P_n(-x) satisfy a Riemann hypothesis. See A142995 for further details. The sequence of values P_n(k) for k = 0,1,2,3, ... produces the n-th row of A108625. (End)
This is the row reversed version of triangle A104684. - Wolfdieter Lang, Sep 12 2016
T(n, k) is also the number of (n-k)-dimensional faces of a convex n-dimensional Lipschitz polytope of real functions f defined on the set X = {1, 2, ..., n+1} which satisfy the condition f(n+1) = 0 (see Gordon and Petrov). - Stefano Spezia, Sep 25 2021
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial ((x+1)*(x+2)*(x+3)*...*(x+n) / n!)^2 in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 09 2022
Chapoton's observation above is correct: the precise expansion is ((x+1)*(x+2)*(x+3)*...*(x+n)/ n!)^2 = Sum_{k = 0..n} (-1)^k*T(n,n-k)*binomial(x+2*n-k, 2*n-k), as can be verified using the WZ algorithm. For example, n = 3 gives ((x+1)*(x+2)*(x+3)/3!)^2 = 20*binomial(x+6,6) - 30*binomial(x+5,5) + 12*binomial(x+4,4) - binomial(x+3,3). - Peter Bala, Jun 24 2023

			The triangle T(n, k) starts:
  n\k 0  1    2     3     4      5      6      7      8     9
  0:  1
  1:  1  2
  2:  1  6    6
  3:  1 12   30    20
  4:  1 20   90   140    70
  5:  1 30  210   560   630    252
  6:  1 42  420  1680  3150   2772    924
  7:  1 56  756  4200 11550  16632  12012   3432
  8:  1 72 1260  9240 34650  72072  84084  51480  12870
  9:  1 90 1980 18480 90090 252252 420420 411840 218790 48620
... reformatted by _Wolfdieter Lang_, Sep 12 2016
From _Petros Hadjicostas_, Jul 11 2020: (Start)
Its inverse (from Table II, p. 92, in Ser's book) is
   1;
  -1/2,  1/2;
   1/3, -1/2,    1/6;
  -1/4,  9/20,  -1/4,   1/20;
   1/5, -2/5,    2/7,  -1/10,  1/70;
  -1/6,  5/14, -25/84,  5/36, -1/28,  1/252;
   1/7, -9/28,  25/84, -1/6,   9/154, -1/84, 1/924;
   ... (End)

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 366.
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, Table I, p. 92.
D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

T. D. Noe, Rows n = 0..100 of triangle, flattened
F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices, arXiv:0809.5123 [math.CO], 2008.
Peter Bala, Deformations of the Hadamard product of power series.
Cyril Banderier, Combinatoire analytique des chemins et des cartes, Thesis (2001), page 49.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009), #09.7.6.
H. J. Brothers, Pascal's Prism: Supplementary Material.
David Callan, A bijection for Delannoy paths, arXiv:2202.04649 [math.CO], 2022.
F. Chapoton, Enumerative properties of generalized associahedra, Séminaire Lotharingien de Combinatoire, B51b (2004), 16 pp.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.
S. Fomin and N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004; arXiv:math/0505518 [math.CO], 2005-2008.
S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15(2) (2002), 497-529.
S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
J. Gordon and F. Petrov, Combinatorics of the Lipschitz Polytope, Arnold Mathematical Journal (2016).
G. Hetyei, Face enumeration using generalized binomial coefficients. This is the draft version of Hetyei's paper referenced below. [Archived version]
Gabor Hetyei, The Stirling polynomial of a simplicial complex Discrete and Computational Geometry 35(3) (2006), 437-455.
Hsien-Kuei Hwang and Satoshi Kuriki, Integrated empirical measures and generalizations of classical goodness-of-fit statistics, arXiv:2404.06040 [math.ST], 2024. See p. 11.
C. Lanczos, Applied Analysis (Annotated scans of selected pages). See page 514.
T. Manneville and V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015.
Thomas Selig, Combinatorial aspects of sandpile models on wheel and fan graphs, arXiv:2202.06487 [math.CO], 2022.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages) See Table I, page 92.
V. Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
R. A. Sulanke, Objects counted by the central Delannoy numbers., J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.
D. Zagier, Integral solutions of Apery-like recurrence equations.

See A331430 for an essentially identical triangle, except with signed entries.
Columns include A000012, A002378, A033487 on the left and A000984, A002457, A002544 on the right.
Main diagonal is A006480.
Row sums are A001850. Alternating row sums are A033999.
Cf. A007318, A008459, A009766, A046899, A104684, A109983, A127674.
Cf. A033282 (f-vectors type A associahedra), A108625, A080721 (f-vectors type D associahedra).
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.)

a063007 n k = a063007_tabl !! n !! k
a063007_row n = a063007_tabl !! n
a063007_tabl = zipWith (zipWith (*)) a007318_tabl a046899_tabl
-- Reinhard Zumkeller, Nov 18 2014

/* As triangle: */ [[Binomial(n,k)*Binomial(n+k,k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 03 2015

p := (n,x) -> orthopoly[P](n,1+2*x): seq(seq(coeff(p(n,x),x,k), k=0..n), n=0..9);

Flatten[Table[Binomial[n, k]Binomial[n + k, k], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Dec 24 2011 *)
Table[CoefficientList[Hypergeometric2F1[-n, n + 1, 1, -x], x], {n, 0, 9}] // Flatten
(* Peter Luschny, Mar 09 2018 *)

{T(n, k) = local(t); if( n<0, 0, t = (x + x^2)^n; for( k=1, n, t=t'); polcoeff(t, k) / n!)} /* Michael Somos, Dec 19 2002 */

{T(n, k) = binomial(n, k) * binomial(n+k, k)} /* Michael Somos, Sep 22 2013 */

{T(n, k) = if( k<0 || k>n, 0, (n+k)! / (k!^2 * (n-k)!))} /* Michael Somos, Sep 22 2013 */

T(n, k) = (n+k)!/(k!^2*(n-k)!) = T(n-1, k)*(n+k)/(n-k) = T(n, k-1)*(n+k)*(n-k+1)/k^2 = T(n-1, k-1)*(n+k)*(n+k-1)/k^2.
binomial(x, n)^2 = Sum_{k>=0} T(n,k) * binomial(x, n+k). - Michael Somos, May 11 2012
T(n, k) = A109983(n, k+n). - Michael Somos, Sep 22 2013
G.f.: G(t, z) = 1/sqrt(1-2*z-4*t*z+z^2). Row generating polynomials = P_n(1+2z), i.e., T(n, k) = [z^k] P_n(1+2*z), where P_n are the Legendre polynomials. - Emeric Deutsch, Apr 20 2004
Sum_{k>=0} T(n, k)*A000172(k) = Sum_{k>=0} T(n, k)^2 = A005259(n). - Philippe Deléham, Jun 08 2005
1 + z*d/dz(log(G(t,z))) = 1 + (1 + 2*t)*z + (1 + 8*t + 8*t^2)*z^2 + ... is the o.g.f. for a signed version of A127674. - Peter Bala, Sep 02 2015
If R(n,t) denotes the n-th row polynomial then x^3 * exp( Sum_{n >= 1} R(n,t)*x^n/n ) = x^3 + (1 + 2*t)*x^4 + (1 + 5*t + 5*t^2)*x^5 + (1 + 9*t + 21*t^2 + 14*t^3)*x^6 + ... is an o.g.f for A033282. - Peter Bala, Oct 19 2015
P(n,x) := 1/(1 + x)*Integral_{t = 0..x} R(n,t) dt are (modulo differences of offset) the row polynomials of A033282. - Peter Bala, Jun 23 2016
From Peter Bala, Mar 09 2018: (Start)
R(n,x) = Sum_{k = 0..n} binomial(2*k,k)*binomial(n+k,n-k)*x^k.
R(n,x) = Sum_{k = 0..n} binomial(n,k)^2*x^k*(1 + x)^(n-k).
n*R(n,x) = (1 + 2*x)*(2*n - 1)*R(n-1,x) - (n - 1)*R(n-2,x).
R(n,x) = (-1)^n*R(n,-1 - x).
R(n,x) = 1/n! * (d/dx)^n ((x^2 + x)^n). (End)
The row polynomials are R(n,x) = hypergeom([-n, n + 1], [1], -x). - Peter Luschny, Mar 09 2018
T(n,k) = C(n+1,k)*A009766(n,k). - Bob Selcoe, Jan 18 2020 (Connects this triangle with the Catalan triangle. - N. J. A. Sloane, Jan 18 2020)
If we let A(n,k) = (-1)^(n+k)*(2*k+1)*(n*(n-1)*...*(n-(k-1)))/((n+1)*...*(n+(k+1))) for n >= 0 and k = 0..n, and we consider both T(n,k) and A(n,k) as infinite lower triangular arrays, then they are inverses of one another. (Empty products are by definition 1.) See the example below. The rational numbers |A(n,k)| appear in Table II on p. 92 in Ser's (1933) book. - Petros Hadjicostas, Jul 11 2020
From Peter Bala, Nov 28 2021: (Start)
Row polynomial R(n,x) = Sum_{k >= n} binomial(k,n)^2 * x^(k-n)/(1+x)^(k+1) for x > -1/2.
R(n,x) = 1/(1 + x)^(n+1) * hypergeom([n+1, n+1], [1], x/(1 + x)).
R(n,x) = (1 + x)^n * hypergeom([-n, -n], [1], x/(1 + x)).
R(n,x) = hypergeom([(n+1)/2, -n/2], [1], -4*x*(1 + x)).
If we set R(-1,x) = 1, we can run the recurrence n*R(n,x) = (1 + 2*x)*(2*n - 1)*R(n-1,x) - (n - 1)*R(n-2,x) backwards to give R(-n,x) = R(n-1,x).
R(n,x) = [t^n] ( (1 + t)*(1 + x*(1 + t)) )^n. (End)
n*T(n,k) = (2*n-1)*T(n-1,k) + (4*n-2)*T(n-1,k-1) - (n-1)*T(n-2,k). - Fabián Pereyra, Jun 30 2022
From Peter Bala, Oct 07 2024: (Start)
n-th row polynomial R(n,x) = Sum_{k = 0..n} binomial(n, k) * x^k o (1 + x)^(n-k), where o denotes the black diamond product of power series as defined by Dukes and White (see Bala, Section 4.4, exercise 3).
Denote this triangle by T. Then T * transpose(T) = A143007, the square array of crystal ball sequences for the A_n X A_n lattices.
Let S denote the triangle ((-1)^(n+k)*T(n, k))n,k >= 0, a signed version of this triangle. Then S^(-1) * T = A007318, Pascal's triangle; it appears that T * S^(-1) = A110098.
T = A007318 * A115951. (End)

A085478 Triangle read by rows: T(n, k) = binomial(n + k, 2*k).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 15, 7, 1, 1, 15, 35, 28, 9, 1, 1, 21, 70, 84, 45, 11, 1, 1, 28, 126, 210, 165, 66, 13, 1, 1, 36, 210, 462, 495, 286, 91, 15, 1, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 1, 55, 495, 1716, 3003, 3003, 1820, 680, 153, 19, 1

Offset: 0

Philippe Deléham, Aug 14 2003

Coefficient array for Morgan-Voyce polynomial b(n,x). A053122 (unsigned) is the coefficient array for B(n,x). Reversal of A054142. - Paul Barry, Jan 19 2004
This triangle is formed from even-numbered rows of triangle A011973 read in reverse order. - Philippe Deléham, Feb 16 2004
T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k+1 peaks. T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k peaks at height >= 2. T(n,k) is the number of directed column-convex polyominoes of area n+1, having k+1 columns. - Emeric Deutsch, May 31 2004
Riordan array (1/(1-x), x/(1-x)^2). - Paul Barry, May 09 2005
The triangular matrix a(n,k) = (-1)^(n+k)*T(n,k) is the matrix inverse of A039599. - Philippe Deléham, May 26 2005
The n-th row gives absolute values of coefficients of reciprocal of g.f. of bottom-line of n-wave sequence. - Floor van Lamoen (fvlamoen(AT)planet.nl), Sep 24 2006
Unsigned version of A129818. - Philippe Deléham, Oct 25 2007
T(n, k) is also the number of idempotent order-preserving full transformations (of an n-chain) of height k >=1 (height(alpha) = |Im(alpha)|) and of waist n (waist(alpha) = max(Im(alpha))). - Abdullahi Umar, Oct 02 2008
A085478 is jointly generated with A078812 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x) = v(1,x) = 1; for n>1, u(n,x) = u(n-1,x)+x*v(n-1)x and v(n,x) = u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Feb 25 2012
Per Kimberling's recursion relations, see A102426. - Tom Copeland, Jan 19 2016
Subtriangle of the triangle given by (0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012
T(n,k) is also the number of compositions (ordered partitions) of 2*n+1 into 2*k+1 parts which are all odd. Proof: The o.g.f. of column k, x^k/(1-x)^(2*k+1) for k >= 0, is the o.g.f. of the odd-indexed members of the sequence with o.g.f. (x/(1-x^2))^(2*k+1) (bisection, odd part). Thus T(n,k) is obtained from the sum of the multinomial numbers A048996 for the partitions of 2*n+1 into 2*k+1 parts, all of which are odd. E.g., T(3,1) = 3 + 3 from the numbers for the partitions [1,1,5] and [1,3,3], namely 3!/(2!*1!) and 3!/(1!*2!), respectively. The number triangle with the number of these partitions as entries is A152157. - Wolfdieter Lang, Jul 09 2012
The matrix elements of the inverse are T^(-1)(n,k) = (-1)^(n+k)*A039599(n,k). - R. J. Mathar, Mar 12 2013
T(n,k) = A258993(n+1,k) for k = 0..n-1. - Reinhard Zumkeller, Jun 22 2015
The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the algebraic function F(x)*G(x)^n about 0, where F(x) = (1 + sqrt(1 + 4*x))/(2*sqrt(1 + 4*x)) and G(x) = ((1 + sqrt(1 + 4*x))/2)^2. For example, for n = 4, (1 + sqrt(1 + 4*x))/(2*sqrt(1 + 4*x)) * ((1 + sqrt(1 + 4*x))/2)^8 = (x^4  + 10*x^3 + 15*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 23 2018
Row n also gives the coefficients of the characteristc polynomial of the tridiagonal n X n matrix M_n given in A332602: Phi(n, x) := Det(M_n - x*1_n) = Sum_{k=0..n} T(n, k)*(-x)^k, for n >= 0, with Phi(0, x) := 1. - Wolfdieter Lang, Mar 25 2020
It appears that the largest root of the n-th degree polynomial is equal to the sum of the distinct diagonals of a (2*n+1)-gon including the edge, 1. The largest root of x^3 - 6*x^2 + 5*x - 1 is 5.048917... = the sum of (1 + 1.80193... + 2.24697...). Alternatively, the largest root of the n-th degree polynomial is equal to the square of sigma(2*n+1). Check: 5.048917... is the square of sigma(7), 2.24697.... Given N = 2*n+1, sigma(N) (N odd) can be defined as 1/(2*sin(Pi/(2*N))). Relating to the 9-gon, the largest root of x^4 - 10*x^3 + 15*x^2 - 7*x + 1 is 8.290859..., = the sum of (1 + 1.879385... + 2.532088... + 2.879385...), and is the square of sigma(9), 2.879385... Refer to A231187 for a further clarification of sigma(7). - Gary W. Adamson, Jun 28 2022
For n >=1, the n-th row is given by the coefficients of the minimal polynomial of -4*sin(Pi/(4*n + 2))^2. - Eric W. Weisstein, Jul 12 2023
Denoting this lower triangular array by L, then L * diag(binomial(2*k,k)^2) * transpose(L) is the LDU factorization of A143007, the square array of crystal ball sequences for the A_n X A_n lattices. - Peter Bala, Feb 06 2024
T(n, k) is the number of occurrences of the periodic substring (01)^k in the periodic string (01)^n (see Proposition 4.7 at page 7 in Fang). - Stefano Spezia, Jun 09 2024

			Triangle begins as:
  1;
  1    1;
  1    3    1;
  1    6    5    1;
  1   10   15    7    1;
  1   15   35   28    9    1;
  1   21   70   84   45   11    1;
  1   28  126  210  165   66   13    1;
  1   36  210  462  495  286   91   15    1;
  1   45  330  924 1287 1001  455  120   17    1;
  1   55  495 1716 3003 3003 1820  680  153   19    1;
...
From _Philippe Deléham_, Mar 26 2012: (Start)
(0, 1, 0, 1, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, ...) begins:
  1
  0, 1
  0, 1,  1
  0, 1,  3,   1
  0, 1,  6,   5,   1
  0, 1, 10,  15,   7,   1
  0, 1, 15,  35,  28,   9,  1
  0, 1, 21,  70,  84,  45, 11,  1
  0, 1, 28, 126, 210, 165, 66, 13, 1. (End)

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - _N. J. A. Sloane_, May 10 2012
Peter Bala, A 4-parameter family of embedded Riordan arrays
E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
Paul Barry, The second production matrix of a Riordan array, arXiv:2011.13985 [math.CO], 2020.
Paul Barry and Aoife Hennessy, Notes on a Family of Riordan Arrays and Associated Integer Hankel Transforms , JIS 12 (2009) 09.5.3.
Eduardo H. M. Brietzke, Recurrence Relations for Sums of Binomial Coefficients and Some Generalizations, Journal of Integer Sequences, Vol. 27 (2024), Article 24.3.4.
E. Czabarka et al, Enumerations of peaks and valleys on non-decreasing Dyck paths, Disc. Math. 341 (2018) 2789-2807, Theorem 3.
Emeric Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
Wenjie Fang, Maximal number of subword occurrences in a word, arXiv:2406.02971 [math.CO], 2024.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62. - _Abdullahi Umar_, Oct 02 2008
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174.
Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, (2005) 359-370.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

Row sums: A001519. Signed versions: A123970, A129818.
Cf. A000108, A001006, A007318, A098158, A183160, A078812, A091042.
Cf. A258993, A025581, A051162, A054142, A332602.
Cf. A231187, A143007.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n+k, 2*k) ))); # G. C. Greubel, Aug 01 2019

a085478 n k = a085478_tabl !! n !! k
a085478_row n = a085478_tabl !! n
a085478_tabl = zipWith (zipWith a007318) a051162_tabl a025581_tabl
-- Reinhard Zumkeller, Jun 22 2015

[Binomial(n+k, 2*k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

T := (n,k) -> binomial(n+k,2*k): seq(seq(T(n,k), k=0..n), n=0..11);

(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A085478 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A078812 *) (*Clark Kimberling, Feb 25 2012 *)
(* Second program *)
Table[Binomial[n + k, 2 k], {n, 0, 12}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 01 2019 *)
CoefficientList[Table[Fibonacci[2 n + 1, Sqrt[x]], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Jul 03 2023 *)
Join[{{1}}, CoefficientList[Table[MinimalPolynomial[-4 Sin[Pi/(4 n + 2)]^2, x], {n, 20}], x]] (* Eric W. Weisstein, Jul 12 2023 *)

PARI
```
T(n,k) = binomial(n+k,n-k)
```

[[binomial(n+k,2*k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

T(n, k) = (n+k)!/((n-k)!*(2*k)!).
G.f.: (1-z)/((1-z)^2-tz). - Emeric Deutsch, May 31 2004
Row sums are A001519 (Fibonacci(2n+1)). Diagonal sums are A011782. Binomial transform of A026729 (product of lower triangular matrices). - Paul Barry, Jun 21 2004
T(n, 0) = 1, T(n, k) = 0 if n=0} T(n-1-j, k-1)*(j+1). T(0, 0) = 1, T(0, k) = 0 if k>0; T(n, k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j). For the column k, g.f.: Sum_{n>=0} T(n, k)*x^n = (x^k) / (1-x)^(2*k+1). - Philippe Deléham, Feb 15 2004
Sum_{k=0..n} T(n,k)*x^(2*k) = A000012(n), A001519(n+1), A001653(n), A078922(n+1), A007805(n), A097835(n), A097315(n), A097838(n), A078988(n), A097841(n), A097727(n), A097843(n), A097730(n), A098244(n), A097733(n), A098247(n), A097736(n), A098250(n), A097739(n), A098253(n), A097742(n), A098256(n), A097767(n), A098259(n), A097770(n), A098262(n), A097773(n), A098292(n), A097776(n) for x=0,1,2,...,27,28 respectively. - Philippe Deléham, Dec 31 2007
T(2*n,n) = A005809(n). - Philippe Deléham, Sep 17 2009
A183160(n) = Sum_{k=0..n} T(n,k)*T(n,n-k). - Paul D. Hanna, Dec 27 2010
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Feb 06 2012
O.g.f. for column k: x^k/(1-x)^(2*k+1), k >= 0. [See the o.g.f. of the triangle above, and a comment on compositions. - Wolfdieter Lang, Jul 09 2012]
E.g.f.: (2/sqrt(x + 4))*sinh((1/2)*t*sqrt(x + 4))*cosh((1/2)*t*sqrt(x)) = t + (1 + x)*t^3/3! + (1 + 3*x + x^2)*t^5/5! + (1 + 6*x + 5*x^2 + x^3)*t^7/7! + .... Cf. A091042. - Peter Bala, Jul 29 2013
T(n, k) = A065941(n+3*k, 4*k) = A108299(n+3*k, 4*k) = A194005(n+3*k, 4*k). - Johannes W. Meijer, Sep 05 2013
Sum_{k=0..n} (-1)^k*T(n,k)*A000108(k) = A000007(n) for n >= 0. - Werner Schulte, Jul 12 2017
Sum_{k=0..floor(n/2)} T(n-k,k)*A000108(k) = A001006(n) for n >= 0. - Werner Schulte, Jul 12 2017
From  Peter Bala, Jun 26 2025: (Start)
The n-th row polynomial b(n, x) = (-1)^n * U(2*n, (i/2)*sqrt(x)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.
b(n, x) = (-1)^n * Dir(n, -1 - x/2), where Dir(n, x) is the n-th row polynomial of the triangle A244419.
b(n, -1 - x) is the n-th row polynomial of A098493. (End)

A081085 Expansion of 1 / AGM(1, 1 - 8*x) in powers of x.

Original entry on oeis.org

1, 4, 20, 112, 676, 4304, 28496, 194240, 1353508, 9593104, 68906320, 500281280, 3664176400, 27033720640, 200683238720, 1497639994112, 11227634469668, 84509490017680, 638344820152784, 4836914483890112, 36753795855173776, 279985580271435584, 2137790149251471680

Offset: 0

Michael Somos, Mar 04 2003

AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre.
This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004
This is the exponential (also known as binomial) convolution of sequence A000984 (central binomial) with itself. See the V. Jovovic e.g.f. and a(n) formulas given below. - Wolfdieter Lang, Jan 13 2012
This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
The recursion (n+1)^2 * a(n+1) = (12*n^2+12*n+4) * a(n) - 32*n^2*a(n-1) with n=0 has zero coefficient for a(-1) and thus a(-1) is not determined uniquely by it, but defining a(-1) = 2^(-5/2) makes a(n) = a(-1-n) * 32^(n-1/2) true for all n in Z. - Michael Somos, Apr 05 2022

			G.f. = A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4304*x^5 + 28496*x^6 + ...

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

Seiichi Manyama, Table of n, a(n) for n = 0..1110 (terms 0..200 from Vincenzo Librandi)
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.
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.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.
E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013-2015.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See E p. 2.
S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.
Xiao-Juan Ji and Zhi-Hong Sun, Congruences for Catalan-Larcombe-French numbers, arXiv:1505.00668 [math.NT], 2015.
Bradley Klee, Checking Weierstrass data, 2023.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
Stéphane Ouvry and Alexios Polychronakos, Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers, arXiv:2006.06445 [math-ph], 2020.
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.
D. Zagier, Integral solutions of Apery-like recurrence equations. See line E in sporadic solutions table of page 5.

Cf. A053175, A089603, A091401.
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.

seq(simplify(binomial(2*n, n)*hypergeom([ -n, -n, 1/2], [1, -n+1/2], -1)), n = 0..22); # Peter Bala, Jul 25 2024

Table[Sum[Binomial[n,k]*Binomial[2*n-2*k,n-k]*Binomial[2*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/2, 1/2, 1, 16 x (1 - 4 x)], {x, 0, n}]; (* Michael Somos, Oct 25 2014 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 1 / NestWhile[ {(#[[1]] + #[[2]])/2, Sqrt[#[[1]] #[[2]]]} &, {1, Series[ 1 - 8 x, {x, 0, n}]}, #[[1]] =!= #[[2]] &] // First, {x, 0, n}]]; (* Michael Somos, Oct 27 2014 *)
CoefficientList[Series[2*EllipticK[1/(1 - 1/(4*x))^2] / (Pi*(1 - 4*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 13 2019 *)
a[n_] := Binomial[2 n, n] HypergeometricPFQ[{1/2, -n, -n},{1, 1/2 - n}, -1];
Table[a[n], {n, 0, 20}] (* Peter Luschny, Apr 05 2022 *)

{a(n) = if( n<0, 0, polcoeff( 1 / agm( 1, 1 - 8 * x + x * O(x^n)), n))};

{a(n) = if( n<0,0, 4^n * sum( k=0, n\2, binomial( n, 2*k) * binomial( 2*k, k)^2 / 16^k))};

{a(n)=n!*polcoeff(sum(k=0,n,(2*k)!*x^k/(k!)^3 +x*O(x^n))^2,n)} /* Paul D. Hanna, Sep 04 2009 */

from math import comb
def A081085(n): return sum((1<<(n-(m:=k<<1)<<1))*comb(n,m)*comb(m,k)**2 for k in range((n>>1)+1)) # Chai Wah Wu, Jul 09 2023

G.f.: 1 / AGM(1, 1 - 8*x).
E.g.f.: exp(4*x)*BesselI(0, 2*x)^2. - Vladeta Jovovic, Aug 20 2003
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*n-2*k, n-k)*binomial(2*k, k) = binomial(2*n, n)*hypergeom([ -n, -n, 1/2], [1, -n+1/2], -1). - Vladeta Jovovic, Sep 16 2003
D-finite with recurrence (n+1)^2 * a(n+1) = (12*n^2+12*n+4) * a(n) - 32*n^2*a(n-1). - Matthijs Coster, Apr 28 2004
E.g.f.: [Sum_{n>=0} binomial(2n,n)*x^n/n! ]^2. - Paul D. Hanna, Sep 04 2009
G.f.: Gaussian Hypergeometric function 2F1(1/2, 1/2; 1; 16*x-64*x^2). - Mark van Hoeij, Oct 24 2011
a(n) = 2^(-n) * A053175(n).
a(n) ~ 2^(3*n+1)/(Pi*n). - Vaclav Kotesovec, Oct 13 2012
0 = x*(x+4)*(x+8)*y'' + (3*x^2 + 24*x + 32)*y' + (x+4)*y, where y(x) = A(x/-32). - Gheorghe Coserea, Aug 26 2016
a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k)^2. - Seiichi Manyama, Apr 02 2017
a(n) = (1/Pi)^2*Integral_{0 <= x, y <= Pi} (4*cos(x)^2 + 4*cos(y)^2)^n dx dy. - Peter Bala, Feb 10 2022
a(n) = a(-1-n)*32^(n-1/2) and 0 = +a(n)*(+a(n+1)*(+32768*a(n+2) -23552*a(n+3) +3072*a(n+4)) +a(n+2)*(-8192*a(n+2) +8448*a(n+3) -1248*a(n+4)) +a(n+3)*(-512*a(n+3) +96*a(n+4))) +a(n+1)*(+a(n+1)*(-5120*a(n+2) +3840*a(n+3) -512*a(n+4)) +a(n+2)*(+1536*a(n+2) -1728*a(n+3) +264*a(n+4)) +a(n+3)*(+120*a(n+3) -23*a(n+4))) +a(n+2)*(+a(n+2)*(-32*a(n+2) +48*a(n+3) -8*a(n+4)) +a(n+3)*(-5*a(n+3) +a(n+4))) for all n in Z. - Michael Somos, Apr 04 2022
From Bradley Klee, Jun 05 2023: (Start)
The g.f. T(x) obeys a period-annihilating ODE:
0=4*(-1 + 8*x)*T(x) + (1 - 24*x + 96*x^2)*T'(x) + x*(-1 + 4*x)*(-1 + 8*x)*T''(x).
The periods ODE can be derived from the following Weierstrass data:
g2 = 3*(1 - 16*(1 - 8*x)^2 + 16*(1 - 8*x)^4);
g3 = 1 + 30*(1 - 8*x)^2 - 96*(1 - 8*x)^4 + 64*(1 - 8*x)^6;
which determine an elliptic surface with four singular fibers. (End)
G.f.: Sum_{n>=0} binomial(2*n,n)^2 * x^n * (1 - 4*x)^n. - Paul D. Hanna, Apr 18 2024
From Peter Bala, Jul 25 2024: (Start)
a(n) = 2*Sum_{k = 1..n} (k/n)*binomial(n, k)*binomial(2*n-2*k, n-k)*binomial(2*k, k) for n >= 1.
a(n-1) = (1/2)*Sum_{k = 1..n} (k/n)^2*binomial(n, k)*binomial(2*n-2*k, n-k)*  binomial(2*k, k) for n >= 1. Cf. A002895. (End)

A108625 Square array, read by antidiagonals, where row n equals the crystal ball sequence for the A_n lattice.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 13, 19, 7, 1, 1, 21, 55, 37, 9, 1, 1, 31, 131, 147, 61, 11, 1, 1, 43, 271, 471, 309, 91, 13, 1, 1, 57, 505, 1281, 1251, 561, 127, 15, 1, 1, 73, 869, 3067, 4251, 2751, 923, 169, 17, 1, 1, 91, 1405, 6637, 12559, 11253, 5321, 1415, 217, 19, 1

Offset: 0

Paul D. Hanna, Jun 12 2005

Compare to the corresponding array A108553 of crystal ball sequences for D_n lattice.
From Peter Bala, Jul 18 2008: (Start)
Row reverse of A099608.
This array has a remarkable relationship with the constant zeta(2). The row, column and diagonal entries of the array occur in series acceleration formulas for zeta(2).
For the entries in row n we have zeta(2) = 2*(1 - 1/2^2 + 1/3^2 - ... + (-1)^(n+1)/n^2) + (-1)^n*Sum_{k >= 1} 1/(k^2*T(n,k-1)*T(n,k)). For example, n = 4 gives zeta(2) = 2*(1 - 1/4 + 1/9 - 1/16) + 1/(1*21) + 1/(4*21*131) + 1/(9*131*471) + ... . See A142995 for further details.
For the entries in column k we have zeta(2) = (1 + 1/4 + 1/9 + ... + 1/k^2) + 2*Sum_{n >= 1} (-1)^(n+1)/(n^2*T(n-1,k)*T(n,k)). For example, k = 4 gives zeta(2) = (1 + 1/4 + 1/9 + 1/16) + 2*(1/(1*9) - 1/(4*9*61) + 1/(9*61*309) - ... ). See A142999 for further details.
Also, as consequence of Apery's proof of the irrationality of zeta(2), we have a series acceleration formula along the main diagonal of the table: zeta(2) = 5 * Sum_{n >= 1} (-1)^(n+1)/(n^2*T(n,n)*T(n-1,n-1)) = 5*(1/3 - 1/(2^2*3*19) + 1/(3^2*19*147) - ...).
There also appear to be series acceleration results along other diagonals. For example, for the main subdiagonal, calculation supports the result zeta(2) = 2 - Sum_{n >= 1} (-1)^(n+1)*(n^2+(2*n+1)^2)/(n^2*(n+1)^2*T(n,n-1)*T(n+1,n)) = 2 - 10/(2^2*7) + 29/(6^2*7*55) - 58/(12^2*55*471) + ..., while for the main superdiagonal we appear to have zeta(2) = 1 + Sum_{n >= 1} (-1)^(n+1)*((n+1)^2 + (2*n+1)^2)/(n^2*(n+1)^2*T(n-1,n)*T(n,n+1)) = 1 + 13/(2^2*5) - 34/(6^2*5*37) + 65/(12^2*37*309) - ... .
Similar series acceleration results hold for Apery's constant zeta(3) involving the crystal ball sequences for the product lattices A_n x A_n; see A143007 for further details. Similar results also hold between the constant log(2) and the crystal ball sequences of the hypercubic lattices A_1 x...x A_1 and between log(2) and the crystal ball sequences for lattices of type C_n ; see A008288 and A142992 respectively for further details. (End)
This array is the Hilbert transform of triangle A008459 (see A145905 for the definition of the Hilbert transform). - Peter Bala, Oct 28 2008

			Square array begins:
  1,   1,    1,     1,      1,       1,       1, ... A000012;
  1,   3,    5,     7,      9,      11,      13, ... A005408;
  1,   7,   19,    37,     61,      91,     127, ... A003215;
  1,  13,   55,   147,    309,     561,     923, ... A005902;
  1,  21,  131,   471,   1251,    2751,    5321, ... A008384;
  1,  31,  271,  1281,   4251,   11253,   25493, ... A008386;
  1,  43,  505,  3067,  12559,   39733,  104959, ... A008388;
  1,  57,  869,  6637,  33111,  124223,  380731, ... A008390;
  1,  73, 1405, 13237,  79459,  350683, 1240399, ... A008392;
  1,  91, 2161, 24691, 176251,  907753, 3685123, ... A008394;
  1, 111, 3191, 43561, 365751, 2181257, ...      ... A008396;
  ...
As a triangle:
  [0]  1
  [1]  1,  1
  [2]  1,  3,   1
  [3]  1,  7,   5,    1
  [4]  1, 13,  19,    7,    1
  [5]  1, 21,  55,   37,    9,    1
  [6]  1, 31, 131,  147,   61,   11,   1
  [7]  1, 43, 271,  471,  309,   91,  13,   1
  [8]  1, 57, 505, 1281, 1251,  561, 127,  15,  1
  [9]  1, 73, 869, 3067, 4251, 2751, 923, 169, 17, 1
       ...
Inverse binomial transform of rows yield rows of triangle A063007:
  1;
  1,  2;
  1,  6,   6;
  1, 12,  30,  20;
  1, 20,  90, 140,  70;
  1, 30, 210, 560, 630, 252; ...
Product of the g.f. of row n and (1-x)^(n+1) generates the symmetric triangle A008459:
  1;
  1,  1;
  1,  4,   1;
  1,  9,   9,   1;
  1, 16,  36,  16,  1;
  1, 25, 100, 100, 25, 1;
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
R. Bacher, P. de la Harpe, and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Joseph T. Iosue, T. C. Mooney, Adam Ehrenberg, and Alexey V. Gorshkov, Projective toric designs, difference sets, and quantum state designs, arXiv:2311.13479 [quant-ph], 2023. See page 6.
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.
A. van der Poorten, A proof that Euler missed ... Apery'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.

Rows include: A003215 (row 2), A005902 (row 3), A008384 (row 4), A008386 (row 5), A008388 (row 6), A008390 (row 7), A008392 (row 8), A008394 (row 9), A008396 (row 10).
Cf. A063007, A099601 (n-th term of A_{2n} lattice), A108553.
Cf. A008459 (h-vectors type B associahedra), A145904, A145905.
Cf. A005258 (main diagonal), A108626 (antidiagonal sums).
Cf. A108628, A208675, A363867, A363868, A363869, A363870, A363871.

T:= func< n,k | (&+[Binomial(n,j)^2*Binomial(n+k-j,k-j): j in [0..k]]) >; // array
A108625:= func< n,k | T(n-k,k) >; // antidiagonals
[A108625(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023

T := (n,k) -> binomial(n, k)*hypergeom([-k, k - n, k - n], [1, -n], 1):
seq(seq(simplify(T(n,k)),k=0..n),n=0..10); # Peter Luschny, Feb 10 2018

T[n_, k_]:= HypergeometricPFQ[{-n, -k, n+1}, {1, 1}, 1] (* Michael Somos, Jun 03 2012 *)

T(n,k)=sum(i=0,k,binomial(n,i)^2*binomial(n+k-i,k-i))

def T(n,k): return sum(binomial(n,j)^2*binomial(n+k-j, k-j) for j in range(k+1)) # array
def A108625(n,k): return T(n-k, k) # antidiagonals
flatten([[A108625(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023

T(n, k) = Sum_{i=0..k} C(n, i)^2 * C(n+k-i, k-i).
G.f. for row n: (Sum_{i=0..n} C(n, i)^2 * x^i)/(1-x)^(n+1).
Sum_{k=0..n} T(n-k, k) = A108626(n) (antidiagonal sums).
From Peter Bala, Jul 23 2008 (Start):
O.g.f. row n: 1/(1 - x)*Legendre_P(n,(1 + x)/(1 - x)).
G.f. for square array: 1/sqrt((1 - x)*((1 - t)^2 - x*(1 + t)^2)) = (1 + x + x^2 + x^3 + ...) + (1 + 3*x + 5*x^2 + 7*x^3 + ...)*t + (1 + 7*x + 19*x^2 + 37*x^3 + ...)*t^2 + ... . Cf. A142977.
Main diagonal is A005258.
Recurrence relations:
Row n entries: (k+1)^2*T(n,k+1) = (2*k^2+2*k+n^2+n+1)*T(n,k) - k^2*T(n,k-1), k = 1,2,3,... ;
Column k entries: (n+1)^2*T(n+1,k) = (2*k+1)*(2*n+1)*T(n,k) + n^2*T(n-1,k), n = 1,2,3,... ;
Main diagonal entries: (n+1)^2*T(n+1,n+1) = (11*n^2+11*n+3)*T(n,n) + n^2*T(n-1,n-1), n = 1,2,3,... .
Series acceleration formulas for zeta(2):
Row n: zeta(2) = 2*(1 - 1/2^2 + 1/3^2 - ... + (-1)^(n+1)/n^2) + (-1)^n*Sum_{k >= 1} 1/(k^2*T(n,k-1)*T(n,k));
Column k: zeta(2) = 1 + 1/2^2 + 1/3^2 + ... + 1/k^2 + 2*Sum_{n >= 1} (-1)^(n+1)/(n^2*T(n-1,k)*T(n,k));
Main diagonal: zeta(2) = 5 * Sum_{n >= 1} (-1)^(n+1)/(n^2*T(n-1,n-1)*T(n,n)).
Conjectural result for superdiagonals: zeta(2) = 1 + 1/2^2 + ... + 1/k^2 + Sum_{n >= 1} (-1)^(n+1) * (5*n^2 + 6*k*n + 2*k^2)/(n^2*(n+k)^2*T(n-1,n+k-1)*T(n,n+k)), k = 0,1,2... .
Conjectural result for subdiagonals: zeta(2) = 2*(1 - 1/2^2 + ... + (-1)^(k+1)/k^2) + (-1)^k*Sum_{n >= 1} (-1)^(n+1)*(5*n^2 + 4*k*n + k^2)/(n^2*(n+k)^2*T(n+k-1,n-1)*T(n+k,n)), k = 0,1,2... .
Conjectural congruences: the main superdiagonal numbers S(n) := T(n,n+1) appear to satisfy the supercongruences S(m*p^r - 1) = S(m*p^(r-1) - 1) (mod p^(3*r)) for all primes p >= 5 and all positive integers m and r. If p is prime of the form 4*n + 1 we can write p = a^2 + b^2 with a an odd number. Then calculation suggests the congruence S((p-1)/2) == 2*a^2 (mod p). (End)
From Michael Somos, Jun 03 2012: (Start)
T(n, k) = hypergeom([-n, -k, n + 1], [1, 1], 1).
T(n, n-1) = A208675(n).
T(n+1, n) = A108628(n). (End)
T(n, k) = binomial(n, k)*hypergeom([-k, k - n, k - n], [1, -n], 1). - Peter Luschny, Feb 10 2018
From Peter Bala, Jun 23 2023: (Start)
T(n, k) = Sum_{i = 0..k} (-1)^i * binomial(n, i)*binomial(n+k-i, k-i)^2.
T(n, k) = binomial(n+k, k)^2 * hypergeom([-n, -k, -k], [-n - k, -n - k], 1). (End)
From Peter Bala, Jun 28 2023; (Start)
T(n,k) = the coefficient of (x^n)*(y^k)*(z^n) in the expansion of 1/( (1 - x - y)*(1 - z ) - x*y*z ).
T(n,k) = B(n, k, n) in the notation of Straub, equation 24.
The supercongruences T(n*p^r, k*p^r) == T(n*p^(r-1), k*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and k.
The formula T(n,k) = hypergeom([n+1, -n, -k], [1, 1], 1) allows the table indexing to be extended to negative values of n and k; clearly, we find that T(-n,k) = T(n-1,k) for all n and k. It appears that T(n,-k) = (-1)^n*T(n,k-1) for n >= 0, while T(n,-k) = (-1)^(n+1)*T(n,k-1) for n <= -1 [added Sep 10 2023: these follow from the identities immediately below]. (End)
T(n,k) = Sum_{i = 0..n} (-1)^(n+i) * binomial(n, i)*binomial(n+i, i)*binomial(k+i, i) = (-1)^n * hypergeom([n + 1, -n, k + 1], [1, 1], 1). - Peter Bala, Sep 10 2023
From G. C. Greubel, Oct 05 2023: (Start)
Let t(n,k) = T(n-k, k) (antidiagonals).
t(n, k) = Hypergeometric3F2([k-n, -k, n-k+1], [1,1], 1).
T(n, 2*n) = A363867(n).
T(3*n, n) = A363868(n).
T(2*n, 2*n) = A363869(n).
T(n, 3*n) = A363870(n).
T(2*n, 3*n) = A363871(n). (End)
T(n, k) = Sum_{i = 0..n} binomial(n, i)*binomial(n+i, i)*binomial(k, i). - Peter Bala, Feb 26 2024
Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*T(n, k) = A005259(n), the Apéry numbers associated with zeta(3). - Peter Bala, Jul 18 2024
From Peter Bala, Sep 21 2024: (Start)
Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*T(n, k) = binomial(2*n, n) = A000984(n).
Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*T(n-1, n-k) = A376458(n).
Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*T(i, k) = A143007(n, i). (End)
From Peter Bala, Oct 12 2024: (Start)
The square array = A063007 * transpose(A007318).
Conjecture: for positive integer m, Sum_{k = 0..n} (-1)^(n+k) * binomial(n, k) * T(m*n, k) = ((m+1)*n)!/( ((m-1)*n)!*n!^2) (verified up to m = 10 using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package). (End)

A006077 (n+1)^2a(n+1) = (9n^2+9n+3)a(n) - 27n^2a(n-1), with a(0) = 1 and a(1) = 3.

Original entry on oeis.org

1, 3, 9, 21, 9, -297, -2421, -12933, -52407, -145293, -35091, 2954097, 25228971, 142080669, 602217261, 1724917221, 283305033, -38852066421, -337425235479, -1938308236731, -8364863310291, -24286959061533, -3011589296289, 574023003011199, 5028616107443691

Offset: 0

N. J. A. Sloane

sign

This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004
Conjecture: Let W(n) be the (n+1) X (n+1) Hankel-type determinant with (i,j)-entry equal to a(i+j) for all i,j = 0,...,n. If n == 1 (mod 3) then W(n) = 0. When n == 0 or 2 (mod 3), W(n)*(-1)^(floor((n+1)/3))/6^n is always a positive odd integer. - Zhi-Wei Sun, Aug 21 2013
Conjecture: Let p == 1 (mod 3) be a prime, and write 4*p = x^2 + 27*y^2 with x, y integers and x == 1 (mod 3). Then W(p-1) == (-1)^{(p+1)/2}*(x-p/x) (mod p^2), where W(n) is defined as the above. - Zhi-Wei Sun, Aug 23 2013
This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
Diagonal of rational functions 1/(1 - (x^2*y + y^2*z - z^2*x + 3*x*y*z)), 1/(1 - (x^3 + y^3 - z^3 + 3*x*y*z)), 1/(1 + x^3 + y^3 + z^3 - 3*x*y*z). - Gheorghe Coserea, Aug 04 2018

			G.f. = 1 + 3*x + 9*x^2 + 21*x^3 + 9*x^4 - 297*x^5 - 2421*x^6 - 12933*x^7 - ...

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

Seiichi Manyama, Table of n, a(n) for n = 0..1400 (terms 0..200 from T. D. Noe)
Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982.
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.
M. Coster, Email, Nov 1990
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See B p. 2.
S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.
Bradley Klee, Checking Weierstrass data, 2023.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
Stéphane Ouvry and Alexios Polychronakos, Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers, arXiv:2006.06445 [math-ph], 2020.
Zhi-Wei Sun, Three mysterious conjectures on Hankel-type determinants, a message to Number Theory List, August 22, 2013.
Zhi-Wei Sun, Connections between p = x^2+3*y^2 and Franel numbers, J. Number Theory 133(2013), 2914-2928.

Related to diagonal of rational functions: A268545-A268555.
Cf. A091401.
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 -> 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], 1):
seq(simplify(a(n)), n=0..24); # Peter Luschny, Nov 01 2017

Table[Sum[(-1)^k*3^(n - 3*k)*Binomial[n, 3*k]*Binomial[2*k, k]* Binomial[3*k, k], {k, 0, Floor[n/3]}], {n, 0, 50}] (* G. C. Greubel, Oct 24 2017 *)
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/3, 2/3}, {1}, x^3 / (x - 1/3)^3 ] / (1 - 3 x), {x, 0, n}]; (* Michael Somos, Nov 01 2017 *)

subst(eta(q)^3/eta(q^3), q, serreverse(eta(q^9)^3/eta(q)^3*q)) \\ (generating function) Helena Verrill (verrill(AT)math.lsu.edu), Apr 20 2009 [for (-1)^n*a(n)]

diag(expr, N=22, var=variables(expr)) = {
  my(a = vector(N));
  for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));
  for (n = 1, N, a[n] = expr;
    for (k = 1, #var, a[n] = polcoeff(a[n], n-1)));
  return(a);
};
diag(1/(1 + x^3 + y^3 + z^3 - 3*x*y*z), 25)

seq(N) = {
  my(a = vector(N)); a[1] = 3; a[2] = 9;
  for (n = 2, N-1, a[n+1] = ((9*n^2+9*n+3)*a[n] - 27*n^2*a[n-1])/(n+1)^2);
  concat(1,a);
};
seq(24)
\\ test: y=subst(Ser(seq(202)), 'x, -'x/27); 0 == x*(x^2+9*x+27)*y'' + (3*x^2+18*x+27)*y' + (x+3)*y
\\ Gheorghe Coserea, Nov 09 2017

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); (-1)^n * polcoeff( subst(eta(x + A)^3 / eta(x^3 + A), x, serreverse( x * eta(x^9 + A)^3 / eta(x + A)^3)), n))}; /* Michael Somos, Nov 01 2017 */

G.f.: hypergeom([1/3, 2/3], [1], x^3/(x-1/3)^3) / (1-3*x). - Mark van Hoeij, Oct 25 2011
a(n) = Sum_{k=0..floor(n/3)}(-1)^k*3^(n-3k)*C(n,3k)*C(2k,k)*C(3k,k). - Zhi-Wei Sun, Aug 21 2013
0 = x*(x^2+9*x+27)*y'' + (3*x^2 + 18*x + 27)*y' + (x + 3)*y, where y(x) = A(x/-27). - Gheorghe Coserea, Aug 26 2016
a(n) = 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], 1). - Peter Luschny, Nov 01 2017
From Bradley Klee, Jun 05 2023: (Start)
The g.f. T(x) obeys a period-annihilating ODE:
0=3*(-1 + 9*x)*T(x) + (-1 + 9*x)^2*T'(x) +  x*(1 - 9*x + 27*x^2)*T''(x).
The periods ODE can be derived from the following Weierstrass data:
g2 = 3*(-8 + 9*(1 - 9*x)^3)*(1 - 9*x);
g3 = 8 - 36*(1 - 9*x)^3 + 27*(1 - 9*x)^6;
which determine an elliptic surface with four singular fibers. (End)

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000

A093388 (n+1)^2a(n+1) = (17n^2+17n+6)a(n) - 72n^2a(n-1).

Original entry on oeis.org

1, 6, 42, 312, 2394, 18756, 149136, 1199232, 9729882, 79527084, 654089292, 5408896752, 44941609584, 375002110944, 3141107339328, 26402533581312, 222635989516122, 1882882811380284, 15967419789558804, 135752058036988848, 1156869080242393644

Offset: 0

Matthijs Coster, Apr 29 2004

nonn

This is the Taylor expansion of a special point on a curve described by Beauville.

This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017

			A(x) = 1 + 6*x + 42*x^2 + 312*x^3 + 2394*x^4 + 18756*x^5 + ... is the g.f.

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

Seiichi Manyama, Table of n, a(n) for n = 0..1050 (terms 0..200 from Vincenzo Librandi)
Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982.
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, May 2011.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.
Matthijs Coster, Sequences
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See F p. 2.
S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.
Bradley Klee, Checking Weierstrass data, 2023.
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012. - From _N. J. A. Sloane_, Dec 16 2012
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.
H. Verrill, Some congruences related to modular forms, Section 2.2.
D. Zagier, Integral solutions of Apery-like recurrence equations. See line F in sporadic solutions table of page 5.

This is the seventh sequence in the family beginning A002894, A006077, A081085, A005258, A000172, A002893.
Cf. A091401.
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.

f:=proc(n) option remember; local m; if n=0 then RETURN(1); fi; if n=1 then RETURN(6); fi; m:=n-1; ((17*m^2+17*m+6)*f(n-1)-72*m^2*f(n-2))/n^2; end;

Table[(-1)^n*Sum[Binomial[n,k]*(-8)^k*Sum[Binomial[n-k,j]^3,{j,0,n-k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

a(n)=(-1)^n*sum(k=0,n,binomial(n,k)*(-8)^k*sum(j=0,n-k,binomial(n-k,j)^3));

seq(N) = {
  my(a = vector(N)); a[1] = 6; a[2] = 42;
  for (n=3, N, a[n] = ((17*n^2 - 17*n + 6)*a[n-1] - 72*(n-1)^2*a[n-2])/n^2);
  concat(1,a);
};
seq(20)  \\ Gheorghe Coserea, Aug 26 2016

a(n) = (-1)^n * Sum_{k=0..n} binomial(n, k) * (-8)^k * Sum_{j=0..n-k} binomial(n-k, j)^3. - Helena Verrill (verrill(AT)math.lsu.edu), Aug 09 2004
G.f.: hypergeom([1/3, 2/3],[1],x^2*(8*x-1)/(2*x-1/3)^3)/(1-6*x). - Mark van Hoeij, Oct 25 2011
a(n) ~ 3^(2*n+3/2)/(Pi*n). - Vaclav Kotesovec, Oct 14 2012
G.f. A(x) satisfies: 0 = x*(x+8)*(x+9)*y'' + (3*x^2 + 34*x + 72)*y' + (x+6)*y, where y(x) = A(-x/72). - Gheorghe Coserea, Aug 26 2016
From Bradley Klee, Jun 05 2023: (Start)
The g.f. T(x) obeys a period-annihilating ODE:
0=6*(-1 + 12*x)*T(x) + (1 - 34*x + 216*x^2)*T'(x) + x*(-1 + 8*x)*(-1 + 9*x)*T''(x).
The periods ODE can be derived from the following Weierstrass data:
g2 = 12*(-1 + 6*x)*(-1 + 18*x - 84*x^2 + 24*x^3);
g3 = -8*(1 - 12*x + 24*x^2)*(-1 + 24*x - 192*x^2 + 504*x^3 + 72*x^4);
which determine an elliptic surface with four singular fibers.  (End)

cp's OEIS Frontend

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

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A002893 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2*k,k).

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A002895 Domb numbers: number of 2n-step polygons on diamond lattice.

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A005260 a(n) = Sum_{k = 0..n} binomial(n,k)^4.

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A063007 T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.

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A006077 (n+1)^2a(n+1) = (9n^2+9n+3)a(n) - 27n^2a(n-1), with a(0) = 1 and a(1) = 3.