A002893 -id:A002893 - cp's OEIS frontend

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

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)

A143007 Square array, read by antidiagonals, where row n equals the crystal ball sequence for the 2*n-dimensional lattice A_n x A_n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 25, 73, 25, 1, 1, 41, 253, 253, 41, 1, 1, 61, 661, 1445, 661, 61, 1, 1, 85, 1441, 5741, 5741, 1441, 85, 1, 1, 113, 2773, 17861, 33001, 17861, 2773, 113, 1, 1, 145, 4873, 46705, 142001, 142001, 46705, 4873, 145, 1

Offset: 0

Peter Bala, Jul 22 2008

The A_n lattice consists of all vectors v = (x_1,...,x_(n+1)) in Z^(n+1) such that x_1 + ... + x_(n+1) = 0. The lattice is equipped with the norm ||v|| = 1/2*(|x_1| + ... + |x_(n+1)|). Pairs of lattice points (v,w) in the product lattice A_n x A_n have norm ||(v,w)|| = ||v|| + ||w||. Then the k-th term in the crystal ball sequence for the A_n x A_n lattice gives the number of such pairs (v,w) for which ||(v,w)|| is less than or equal to k.
This array has a remarkable relationship with Apery's constant zeta(3). The row (or column) and main diagonal entries of the array occur in series acceleration formulas for zeta(3). For row n entries there holds zeta(3) = (1+1/2^3+...+1/n^3) + Sum_{k >= 1} 1/(k^3*T(n,k-1)*T(n,k)). Also, as consequence of Apery's proof of the irrationality of zeta(3), we have a series acceleration formula along the main diagonal of the table: zeta(3) = 6 * sum {n >= 1} 1/(n^3*T(n-1,n-1)*T(n,n)). Apery's result appears to generalize to the other diagonals of the table. Calculation suggests the following result may hold: zeta(3) = 1 + 1/2^3 + ... + 1/k^3 + Sum_{n >= 1} (2*n+k)*(3*n^2 +3*n*k +k^2)/(n^3*(n+k)^3*T(n-1,n+k-1)*T(n,n+k)).
For the corresponding results for the constant zeta(2), related to the crystal ball sequences of the lattices A_n, see A108625. For corresponding results for log(2), coming from either the crystal ball sequences of the hypercubic lattices A_1 x ... x A_1 or the lattices of type C_n, see A008288 and A142992 respectively.

			The table begins
n\k|0...1.....2......3.......4.......5
======================================
0..|1...1.....1......1.......1.......1
1..|1...5....13.....25......41......61 A001844
2..|1..13....73....253.....661....1441 A143008
3..|1..25...253...1445....5741...17861 A143009
4..|1..41...661...5741...33001..142001 A143010
5..|1..61..1441..17861..142001..819005 A143011
........
Example row 1 [1,5,13,...]:
The lattice A_1 x A_1 is equivalent to the square lattice of all integer lattice points v = (x,y) in Z x Z equipped with the taxicab norm ||v|| = (|x| + |y|). There are 4 lattice points (marked with a 1 on the figure below) satisfying ||v|| = 1 and 8 lattice points (marked with a 2 on the figure) satisfying ||v|| = 2. Hence the crystal ball sequence for the A_1 x A_1 lattice begins 1, 1+4 = 5, 1+4+8 = 13, ... .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . 2 . . . . .
. . . . 2 1 2 . . . .
. . . 2 1 0 1 2 . . .
. . . . 2 1 2 . . . .
. . . . . 2 . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
Row 1 = [1,5,13,...] is the sequence of partial sums of A008574; row 2 = [1,13,73,...] is the sequence of partial sums of A008530, so row 2 is the crystal ball sequence for the lattice A_2 x A_2 (the 4-dimensional di-isohexagonal orthogonal lattice).
Read as a triangle the array begins
n\k|0...1....2....3...4...5
===========================
0..|1
1..|1...1
2..|1...5....1
3..|1..13...13....1
4..|1..25...73...25...1
5..|1..41..253..253..41...1

G. C. Greubel, Antidiagonals n = 0..50, 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.
J. H. Conway and N. J. A. Sloane, Low dimensional lattices VII Coordination sequences, Proc. R. Soc. Lond., Ser. A, 453 (1997), 2369-2389.
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.

Cf. A001844 (row 1), A005259 (main diagonal), A008288, A008530 (first differences of row 2), A008574 (first differences of row 1), A085478, A108625, A142992, A143003, A143004, A143005, A143006, A143008 (row 2), A143009 (row 3), A142010 (row 4), A143011 (row 5).
Cf. A227845 (antidiagonal sums), A246464.
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.)
Cf. A246563, A352653.

A:= func< n,k | (&+[(Binomial(n,j)*Binomial(n+k-j,k-j))^2: j in [0..n]]) >; // Array
A143007:= func< n,k | A(n-k,k) >; // Antidiagonal triangle
[A143007(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023

with(combinat): T:= (n,k) -> add(binomial(n+j,2*j)*binomial(2*j,j)^2*binomial(k+j,2*j), j = 0..n): for n from 0 to 9 do seq(T(n,k),k = 0..9) end do;

T[n_, k_]:= HypergeometricPFQ[{-k, k+1, -n, n+1}, {1, 1, 1}, 1]; Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Mar 06 2013 *)

/* Print as a square array: */
{T(n, k)=sum(j=0, n, binomial(n+j, 2*j)*binomial(2*j, j)^2*binomial(k+j, 2*j))}
for(n=0, 10, for(k=0,10, print1(T(n,k), ", "));print(""))

/* (1) G.f. A(x,y) when read as a triangle: */
{T(n,k)=local(A=1+x); A=sum(m=0, n, x^m * y^m / (1-x +x*O(x^n))^(2*m+1) * sum(k=0, m, binomial(m, k)^2*x^k)^2 ); polcoeff(polcoeff(A, n,x), k,y)}
for(n=0, 10, for(k=0,n, print1(T(n,k), ", "));print(""))

/* (2) G.f. A(x,y) when read as a triangle: */
{T(n,k)=local(A=1+x); A=sum(m=0, n, x^m/(1-x*y +x*O(x^n))^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * x^k * y^k)^2 ); polcoeff(polcoeff(A, n,x), k,y)}
for(n=0, 10, for(k=0,n, print1(T(n,k), ", "));print(""))

/* (3) G.f. A(x,y) when read as a triangle: */
{T(n,k)=local(A=1+x); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m , k)^2 * y^k * sum(j=0, k, binomial(k, j)^2 * x^j)+x*O(x^n))); polcoeff(polcoeff(A, n,x), k,y)}
for(n=0, 10, for(k=0,n, print1(T(n,k), ", "));print(""))

/* (4) G.f. A(x,y) when read as a triangle: */
{T(n,k)=local(A=1+x); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * y^(m-k) * sum(j=0, k, binomial(k, j)^2 * x^j * y^j)+x*O(x^n))); polcoeff(polcoeff(A, n,x), k,y)}
for(n=0, 10, for(k=0,n, print1(T(n,k), ", "));print(""))
/* End */

def A(n,k): return sum((binomial(n,j)*binomial(n+k-j,k-j))^2 for j in range(n+1)) # array
def A143007(n,k): return A(n-k,k) # antidiagonal triangle
flatten([[A143007(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023

T(n,k) = Sum_{j = 0..n} C(n+j,2*j)*C(2*j,j)^2*C(k+j,2*j).
The array is symmetric T(n,k) = T(k,n).
The main diagonal [1,5,73,1445,...] is the sequence of Apery numbers A005259.
The entries in the k-th column satisfy the Apery-like recursion n^3*T(n,k) + (n-1)^3*T(n-2,k) = (2*n-1)*(n^2-n+1+2*k^2+2*k)*T(n-1,k).
The LDU factorization of the square array is L * D * transpose(L), where L is the lower triangular array A085478 and D is the diagonal matrix diag(C(2n,n)^2). O.g.f. for row n: The generating function for the coordination sequence of the lattice A_n is [Sum_{k = 0..n} C(n,k)^2*x^k ]/(1-x)^n. Thus the generating function for the coordination sequence of the product lattice A_n x A_n is {[Sum_{k = 0..n} C(n,k)^2*x^k]/(1-x)^n}^2 and hence the generating function for row n of this array, the crystal ball sequence of the lattice A_n x A_n, equals [Sum_{k = 0..n} C(n,k)^2*x^k]^2/(1-x)^(2n+1) = 1/(1-x)*[Legendre_P(n,(1+x)/(1-x))]^2. See [Conway & Sloane].
Series acceleration formulas for zeta(3): Row n: zeta(3) = (1 + 1/2^3 + ... + 1/n^3) + Sum_{k >= 1} 1/(k^3*T(n,k-1)*T(n,k)), n = 0,1,2,... . For example, the fourth row of the table (n = 3) gives zeta(3) = (1 + 1/2^3 + 1/3^3) + 1/(1^3*1*25) + 1/(2^3*25*253) + 1/(3^3*253*1445) + ... . See A143003 for further details.
Main diagonal: zeta(3) = 6 * Sum_{n >= 1} 1/(n^3*T(n-1,n-1)*T(n,n)). Conjectural result for other diagonals: zeta(3) = 1 + 1/2^3 + ... + 1/k^3 + Sum_{n >= 1} (2*n+k)*(3*n^2+3*n*k+k^2)/(n^3*(n+k)^3*T(n-1,n+k-1)*T(n,n+k)).
Sum_{k=0..n} T(n-k,k) = A227845(n) (antidiagonal sums). - Paul D. Hanna, Aug 27 2014
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 prime p >= 5 and m, r in N (this is true: see A352653. - Peter Bala, Apr 16 2022).
From Paul D. Hanna, Aug 27 2014: (Start)
G.f. A(x,y) = Sum_{n>=0, k=0..n} T(n,k)*x^n*y^k can be expressed by:
(1) Sum_{n>=0} x^n * y^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k]^2,
(2) Sum_{n>=0} x^n / (1 - x*y)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k * y^k]^2,
(3) Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * y^k * Sum_{j=0..k} C(k,j)^2 * x^j,
(4) Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * y^(n-k) * Sum_{j=0..k} C(k,j)^2 * x^j * y^j. (End)
From Peter Bala, Jun 23 2023: (Start)
T(n,k) = Sum_{j = 0..n} C(n,j)^2 * C(n+k-j,k-j)^2.
T(n,k) = binomial(n+k,k)^2 * hypergeom([-n, -n, -k, -k],[-n - k, -n - k, 1], 1).
T(n,k) = hypergeom([n+1, -n, k+1, -k], [1, 1, 1], 1). (End)
From Peter Bala, Jun 28 2023: (Start)
T(n,k) = the coefficient of (x*z)^n*(y*t)^k in the expansion of 1/( (1 - x - y)*(1 - z - t) - x*y*z*t ).
T(n,k) = A(n, k, n, k) in the notation of Straub, equation 7.
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, -k], [1, 1, 1], 1) allows the table indexing to be extended to negative values of n and k; we have T(-n,k) = T(n-1,k) and T(n,-k) = T(n,k-1) leading to T(-n,-k) = T(n-1, k-1). (End)
From G. C. Greubel, Oct 05 2023: (Start)
Let t(n, k) = T(n-k, k) be the antidiagonal triangle, then:
t(n, k) = t(n, n-k).
Sum_{k=0..floor(n/2)} t(n-k,k) = A246563(n).
t(2*n+1, n+1) = A352653(n+1). (End)
From Peter Bala, Sep 27 2024: (Start)
The square array = A063007 * transpose(A063007) (LU factorization).
Let L denote the lower triangular array (l(n,k))n,k >= 0, where l(n, k) = (-1)^(n+k) * binomial(n, k)*binomial(n+k, k). (L is a signed version of A063007 and L = A063007 * A007318 ^(-1).)
Then the square array = L * transpose(A108625).
L^2 * transpose(A108625) = the Hadamard product of A108625 with itself (both identities can be verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package to find recurrences for the double sums involved). (End)

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A026375 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2k,k).

Original entry on oeis.org

1, 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605, 23973456915, 116312293305, 565280386625, 2751474553575, 13411044301945, 65448142561035, 319756851757695

Offset: 0

Clark Kimberling

nonn

a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=0; also a(n)=T(2n,n).
Partial sums of A085362. Number of bilateral Schroeder paths (i.e., lattice paths consisting of steps U=(1,1), D=(1,-1) and H=(2,0)) from (0,0) to (2n,0) and with no H-steps at odd (positive or negative) levels. Example: a(2)=11 because we have HUD, UDH, UDUD, UUDD, UDDU, their reflections in the x-axis and HH. - Emeric Deutsch, Jan 30 2004
Largest coefficient of (1+3*x+x^2)^n; row sums of triangle in A124733. - Philippe Deléham, Oct 02 2007
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in three colors. - N-E. Fahssi, Feb 05 2008
Equals INVERT transform of A109033: (1, 2, 6, 22, 88, ...), INVERTi transform of A111966, binomial transform of A000984, and inverse Binomial transform of A081671. Convolved with A002212: (1, 3, 10, 36, ...) = A026376: (1, 6, 30, 144, ...). Equals convolution square root of A003463: (1, 6, 31, 156, 781, 3906, ...). - Gary W. Adamson, May 17 2009
Diagonal of array with rational generating function 1/(1 - (x^2 + 3*x*y + y^2)). - Gheorghe Coserea, Jul 29 2018
a(n) == 0 (mod 3) if and only if n is in A081606. - Fabio Visonà, Aug 03 2023

			G.f. = 1 + 3*x + 11*x^2 + 45*x^3 + 195*x^4 + 873*x^5 + 3989*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
David Callan, A combinatorial interpretation for an identity of Barrucand, JIS 11 (2008) 08.3.4.
Shu-Chiuan Chang and Robert Shrock, Structure of the Partition Function and transfer matrices for the Potts model in a magnetic field on lattice strips, J. Stat. Phys. 137 (2009) 667.
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Rui Duarte and António Guedes de Oliveira, Generating functions of lattice paths, Univ. do Porto (Portugal 2023).
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012.
Francesc Fite and Andrew V. Sutherland, Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1, arXiv preprint arXiv:1203.1476 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 14 2012
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Mathematics Stack Exchange, Proof that a(n) == 0 (mod 3) if and only if n is in A081606.
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.
H. D. Nguyen and D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.

Column 3 of A292627. Column 1 of A110165. Central column of A272866.
Cf. A002893, A084610, A000172, A002212.
First differences are in A085362. Bisection of A026380.
m-th binomial transforms of A000984: A126869 (m = -2), A002426 (m = -1 and m = -3 for signed version), A000984 (m = 0 and m = -4 for signed version), A026375 (m = 1 and m = -5 for signed version), A081671 (m = 2 and m = -6 for signed version), A098409 (m = 3 and m = -7 for signed version), A098410 (m = 4 and m = -8 for signed version), A104454 (m = 5 and m = -9 for signed version).

List([0..25],n->Sum([0..n],k->Binomial(n,k)*Binomial(2*k,k))); # Muniru A Asiru, Jul 29 2018

a026375 n = a026374 (2 * n) n  -- Reinhard Zumkeller, Feb 22 2014

seq( add(binomial(n,k)*binomial(2*k,k), k=0..n), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001
a := n -> simplify(GegenbauerC(n, -n, -3/2)):
seq(a(n), n=0..22); # Peter Luschny, May 09 2016

Table[SeriesCoefficient[1/Sqrt[1-6*x+5*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
(* From Michael Somos, May 11 2014: (Start) *)
a[ n_] := Sum[ Binomial[n, k] Binomial[2 k, k], {k, 0, n}];
a[ n_] := If[ n < 0, 0, Hypergeometric2F1[-n, 1/2, 1, -4]];
a[ n_] := If[ n < 0, 0, Coefficient[(1 + 3 x + x^2)^n, x, n]];
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[Exp[3 x] BesselI[0,2 x], {x, 0, n}]];
(* (End) *)

A026375(n):=coeff(expand((1+3*x+x^2)^n),x,n);
makelist(A026375(n),n,0,12); /* Emanuele Munarini, Mar 02 2011 */

{a(n) = if( n<0, 0, polcoeff( (1 + 3*x + x^2)^n, n))}; /* Michael Somos, Sep 09 2002 */

a(n)={my(v=Vec((1-x-x^2)^n)); sum(k=1,#v, v[k]^2);} \\ Joerg Arndt, Jul 06 2011

{a(n) = sum(k=0, n, 5^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019

{a(n) = sum(k=0, n\2, 3^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

Representation by Gauss's hypergeometric function, in Maple notation: a(n)=hypergeom([ -n, 1/2 ], [ 1 ], -4). - Karol A. Penson, Apr 20 2001
This sequence is the binomial transform of A000984. - John W. Layman, Aug 11 2000; proved by Emeric Deutsch, Oct 26 2002
E.g.f.: exp(3*x)*I_0(2x), where I_0 is Bessel function. - Michael Somos, Sep 17 2002
G.f.: 1/sqrt(1-6*x+5*x^2). - Emeric Deutsch, Oct 26 2002
D-finite with recurrence: n*a(n)-3*(2*n-1)*a(n-1)+5*(n-1)*a(n-2)=0 for n > 1. - Emeric Deutsch, Jan 24 2004
From Emeric Deutsch, Jan 30 2004: (Start)
a(n) = [t^n](1+3*t+t^2)^n;
a(n) = Sum_{j=ceiling(n/2)..n} 3^(2*j-n)*binomial(n, j)*binomial(j, n-j). (End)
a(n) = A026380(2*n-1) (n>0). - Emeric Deutsch, Feb 18 2004
G.f.: 1/(1-x-2*x/(1-x/(1-x-x/(1-x/(1-x-x/(1-x/(1-x-x/(1-x... (continued fraction). - Paul Barry, Jan 06 2009
a(n) = sum of squared coefficients of (1+x-x^2)^n - see triangle A084610. - Paul D. Hanna, Jul 18 2009
a(n) = sum of squares of coefficients of (1-x-x^2)^n. - Joerg Arndt, Jul 06 2011
a(n) = (1/Pi)*Integral_{x=-2..2} ((3+x)^n/sqrt((2-x)*(2+x))) dx. - Peter Luschny, Sep 12 2011
a(n) ~ 5^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
G.f.: G(0)/(1-x), where G(k) = 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013
0 = a(n)*(+25*a(n+1) - 45*a(n+2) + 10*a(n+3)) + a(n+1)*(-15*a(n+1) + 36*a(n+2) - 9*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, May 11 2014
a(n) = GegenbauerC(n, -n, -3/2). - Peter Luschny, May 09 2016
a(n) = Sum_{k=0..n} 5^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019
a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019
a(n) = (1/Pi) * Integral_{x = -1..1} (1 + 4*x^2)^n/sqrt(1 - x^2) dx = (1/Pi) * Integral_{x = -1..1} (5 - 4*x^2)^n/sqrt(1 - x^2) dx. - Peter Bala, Jan 27 2020
From Peter Bala, Jan 10 2022: (Start)
1 + x*exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 3*x^2 + 10*x^3 + 36*x^4 + ... is the o.g.f. of A002212.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

Definition simplified by N. J. A. Sloane, Feb 16 2012

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)

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)

A125143 Almkvist-Zudilin numbers: Sum_{k=0..n} (-1)^(n-k) * ((3^(n-3k) (3k)!) / (k!)^3) binomial(n,3k) binomial(n+k,k).

Original entry on oeis.org

1, -3, 9, -3, -279, 2997, -19431, 65853, 292329, -7202523, 69363009, -407637387, 702049401, 17222388453, -261933431751, 2181064727997, -10299472204311, -15361051476987, 900537860383569, -10586290198314843, 74892552149042721, -235054958584593843

Offset: 0

R. K. Guy, Jan 11 2007

Apart from signs, this is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
Diagonal of rational function 1/(1 - (x + y + z + w - 27*x*y*z*w)). - Gheorghe Coserea, Oct 14 2018
Named after the Swedish mathematician Gert Einar Torsten Almkvist (1934-2018) and the Russian mathematician Wadim Walentinowitsch Zudilin (b. 1970). - Amiram Eldar, Jun 23 2021

G. Almkvist and W. Zudilin, Differential equations, mirror maps and zeta values. In Mirror Symmetry V, N. Yui, S.-T. Yau, and J. D. Lewis (eds.), AMS/IP Studies in Advanced Mathematics 38 (2007), International Press and Amer. Math. Soc., pp. 481-515. Cited in Chan & Verrill.
Helena Verrill, in a talk at the annual meeting of the Amer. Math. Soc., New Orleans, LA, Jan 2007 on "Series for 1/pi".

Seiichi Manyama, Table of n, a(n) for n = 0..1052 (terms 0..200 from Arkadiusz Wesolowski)
Gert Almkvist, Christian Krattenthaler, and Joakim Petersson, Some new formulas for pi, Experiment. Math., Vol. 12 (2003), pp. 441-456. (Math Rev MR2043994 by W. Zudilin)
G. Almkvist and W. Zudilin, Differential equations, mirror maps and zeta values, arXiv:math/0402386 [math.NT], 2004.
Tewodros Amdeberhan, and Roberto Tauraso, Supercongruences for the Almkvist-Zudilin numbers, arXiv:1506.08437 [math.NT], 2015.
Yuliy Baryshnikov, Stephen Melczer, Robin Pemantle and Armin Straub, Diagonal asymptotics for symmetric rational functions via ACSV, LIPIcs Proceedings of Analysis of Algorithms 2018, arXiv:1804.10929 [math.CO], 2018.
Heng Huat Chan and Helena Verrill, The Apery numbers, the Almkvist-Zudilin numbers and new series for 1/Pi, Math. Res. Lett., Vol. 16, No. 3 (2009), pp. 405-420.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023. See Table 2 p. 7.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See delta p. 3.
Ji-Cai Liu, A p-adic analogue of Chan and Verrill's formula for 1/Pi, arXiv:2008.06675 [math.NT], 2020.
Ji-Cai Liu, A supercongruence related to Ramanujan-type formula for 1/Pi, Bull. Math. Soc. Sci. Math. Roumanie Tome 67 (115), No. 4, 2024, 483-492. See p. 483.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, Vol. 2, (2016), Article 5.
Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.
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.

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.

Table[Sum[(-1)^(n-k)*((3^(n-3*k)*(3*k)!)/(k!)^3) *Binomial[n,3*k] *Binomial[n+k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 11 2013 *)

a(n) = sum(k=0,n, (-1)^(n-k)*((3^(n-3*k)*(3*k)!)/(k!)^3)*binomial(n,3*k)*binomial(n+k,k) );

a(n) = Sum_{k=0..n} (-1)^(n-k) * ((3^(n-3*k) * (3*k)!) / (k!)^3) * binomial(n,3*k) * binomial(n+k,k) . - Arkadiusz Wesolowski, Jul 13 2011
Recurrence: n^3*a(n) = -(2*n-1)*(7*n^2 - 7*n + 3)*a(n-1) - 81*(n-1)^3*a(n-2). - Vaclav Kotesovec, Sep 11 2013
Lim sup n->infinity |a(n)|^(1/n) = 9. - Vaclav Kotesovec, Sep 11 2013
G.f. y=A(x) satisfies: 0 = x^2*(81*x^2 + 14*x + 1)*y''' + 3*x*(162*x^2 + 21*x + 1)*y'' + (21*x + 1)*(27*x + 1)*y' + 3*(27*x + 1)*y. - Gheorghe Coserea, Oct 15 2018
G.f.: hypergeom([1/8, 5/8], [1], -256*x^3/((81*x^2 + 14*x + 1)*(-x + 1)^2))^2/((81*x^2 + 14*x + 1)^(1/4)*sqrt(-x + 1)). - Sergey Yurkevich, Aug 31 2020

Edited and more terms added by Arkadiusz Wesolowski, Jul 13 2011

A229111 Expansion of the g.f. of A053723 in powers of the g.f. of A121591.

Original entry on oeis.org

1, -5, 35, -275, 2275, -19255, 163925, -1385725, 11483875, -91781375, 688658785, -4581861025, 22550427925, 8852899375, -2431720493125, 47471706909725, -699843878180125, 9141002535744625, -111232778205154375, 1288777160650004375, -14372445132730778975

Offset: 1

Michael Somos, Sep 30 2013

sign

In Verrill (1999) section 2.1, t = (eta(q^5) / eta(q))^6 the g.f. of A121591 and f = eta(q^5)^5 / eta(q) the g.f. of A053723.

Apart from signs, this is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017

			G.f. = x - 5*x^2 + 35*x^3 - 275*x^4 + 2275*x^5 - 19255*x^6 + 163925*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..958
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023. See Table 2 p. 7.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See eta p. 3.
L. O'Brien, Modular forms and two new integer sequences at level 7, Massey University, 2016.
H. Verrill, Some Congruences related to modular forms, Max Planck Institute, 1999.

Cf. A053723, A109064, A121591.
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_] := a[n] = Switch[n, 1, 1, 2, -5, _, (1/(n-1)^3) ((1-2(n-1)) (11(n-2) (n-1)+5) a[n-1] - 125 (n-2)^3 a[n-2])];
a /@ Range[21] (* Jean-François Alcover, Jan 13 2020 *)

{a(n) = my(m = n-1); if( n<1, 0, if( n<3, [1, -5][n], -( (5*(m - 1))^3*a(n-2) + (2*m - 1)*(11*(m^2 - m) +5)*a(n-1) )/ m^3))};

{a(n) = sum(k=0, n-1, (-1)^k*binomial(n-1, k)^3*binomial(5*k-(n-1), 3*(n-1)))} \\ Seiichi Manyama, Sep 02 2020

n^3 * a(n+1) = -(2*n - 1)*(11*n*(n - 1) + 5) * a(n) - 125 * (n - 1)^3 * a(n-1).
a(n*p^k) == (p^3 + Kronecker(p, 5)) * a(n*p^(k-1)) - Kronecker(p, 5) * p^3*a(n*p^(-2)) (mod p^k). [Verrill, 1999]
a(n) = Sum_{k=0..n-1} (-1)^k * binomial(n-1,k)^3 * binomial(5*k-(n-1),3*(n-1)). - Seiichi Manyama, Sep 02 2020

cp's OEIS Frontend

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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A143007 Square array, read by antidiagonals, where row n equals the crystal ball sequence for the 2*n-dimensional lattice A_n x A_n.

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A026375 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k).

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A026375 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2k,k).

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

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

A125143 Almkvist-Zudilin numbers: Sum_{k=0..n} (-1)^(n-k) * ((3^(n-3k) (3k)!) / (k!)^3) binomial(n,3k) binomial(n+k,k).