A169714 -id:A169714 - cp's OEIS frontend

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

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Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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

A287316 Square array A(n,k) = (n!)^2 [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 20, 1, 0, 1, 5, 28, 93, 70, 1, 0, 1, 6, 45, 256, 639, 252, 1, 0, 1, 7, 66, 545, 2716, 4653, 924, 1, 0, 1, 8, 91, 996, 7885, 31504, 35169, 3432, 1, 0, 1, 9, 120, 1645, 18306, 127905, 387136, 272835, 12870, 1, 0

Offset: 0

Peter Luschny, May 23 2017

A287314 provide polynomials and A287315 rational functions generating the columns of the array.

			Arrays start:
k\n| 0  1    2      3         4        5          6           7
---|----------------------------------------------------------------
k=0| 1, 0,   0,      0,       0,       0,         0,          0, ... A000007
k=1| 1, 1,   1,      1,       1,       1,         1,          1, ... A000012
k=2| 1, 2,   6,     20,      70,     252,       924,       3432, ... A000984
k=3| 1, 3,  15,     93,     639,    4653,     35169,     272835, ... A002893
k=4| 1, 4,  28,    256,    2716,   31504,    387136,    4951552, ... A002895
k=5| 1, 5,  45,    545,    7885,  127905,   2241225,   41467725, ... A169714
k=6| 1, 6,  66,    996,   18306,  384156,   8848236,  218040696, ... A169715
k=7| 1, 7,  91,   1645,   36715,  948157,  27210169,  844691407, ...
k=8| 1, 8, 120,   2528,   66424, 2039808,  70283424, 2643158400, ... A385286
k=9| 1, 9, 153,   3681,  111321, 3965409, 159700401, 7071121017, ...
       A000384,A169711, A169712, A169713,                            A033935

Jeremy Tan, Antidiagonals n = 0..50, flattened
Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See p. 19.
Ryan S. Bennink, Counting Abelian Squares for a Problem in Quantum Computing, arXiv:2208.02360 [quant-ph], 2022.
Jonathan M. Borwein, A short walk can be beautiful, preprint, Journal of Humanistic Mathematics, Volume 6 Issue 1 (January 2016), pages 86-109.
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals, preprint, Theoretical Computer Science, Volume 479, 1 April 2013, Pages 4-21.
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, preprint, FPSAC 2010, San Francisco, USA.
L. Bruce Richmond and Jeffrey Shallit, Counting Abelian Squares, 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.

Rows: A000007 (k=0), A000012 (k=1), A000984 (k=2), A002893 (k=3), A002895 (k=4), A169714 (k=5), A169715 (k=6), A385286 (k=8).
Columns: A001477(n=1), A000384 (n=2), A169711 (n=3), A169712 (n=4), A169713 (n=5).
Cf. A033935 (diagonal), A287314, A287315, A287318.

A287316_row := proc(k, len) local b, ser;
b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);
seq((i!)^2*coeff(ser,x,i), i=0..len-1) end:
for k from 0 to 6 do A287316_row(k, 9) od;
A287316_col := proc(n, len) local k, x;
sum(z^k/k!^2, k = 0..infinity); series(%^x, z=0, n+1):
unapply(n!^2*coeff(%, z, n), x); seq(%(j), j=0..len) end:
for n from 0 to 7 do A287316_col(n, 9) od;

Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (n!)^2, {n, 0, 6}], {k, 0,9}]

A287316_row(K, N) = {
  my(x='x + O('x^(2*N-1)));
  Vec(serlaplace(serlaplace(substpol(besseli(0,2*x)^K, 'x^2, 'x))));
};
N=8; concat([vector(N, n, n==1)], vector(9, k, A287316_row(k, N))) \\ Gheorghe Coserea, Jan 12 2018

{A(n, k) = if(n<0 || k<0, 0, n!^2 * polcoeff(besseli(0, 2*x + x*O(x^(2*n)))^k, 2*n))}; /* Michael Somos, Dec 30 2021 */

A(k, n) = my(x='x+O('x^(n+1))); n!^2*polcoeff(hypergeom([], [1], x)^k, n); \\ Peter Luschny, Jun 24 2025

from math import comb
from functools import lru_cache
@lru_cache(maxsize=None)
def A(n,k):
    if k <= 0: return 0**n
    return sum(A(i,k-1)*comb(n,i)**2 for i in range(n+1))
for k in range(10): print([A(n, k) for n in range(8)])
# Jeremy Tan, Dec 10 2021

A(n,k) = A287318(n,k) / binomial(2*n,n).

If a+b=k then A(n,k) = Sum_{i=0..n} A(i,a)*A(n-i,b)*binomial(n,i)^2 (Richmond and Shallit). In particular A(n,k) = Sum_{i=0..n} A(i,k-1)*binomial(n,i)^2. - Jeremy Tan, Dec 10 2021

A169715 The function W_6(2n) (see Borwein et al. reference for definition).

Original entry on oeis.org

1, 6, 66, 996, 18306, 384156, 8848236, 218040696, 5651108226, 152254667436, 4229523740916, 120430899525096, 3499628148747756, 103446306284890536, 3102500089343886696, 94219208840385966096, 2892652835496484004226, 89662253086458906345036

Offset: 0

N. J. A. Sloane, Apr 17 2010

nonn

Row sums of the fifth power of A008459. - Peter Bala, Mar 05 2013

a(n)/6^(2n) is the probability that two throws of n 6-sided dice will give the same result - Henry Bottomley, Aug 30 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Bernstein and T. Lange, Two grumpy giants and a baby, in ANTS X, Proc. Tenth Algorithmic Number Theory Symposium, 2013.
J. M. Borwein, A short walk can be beautiful, preprint, Journal of Humanistic Mathematics, Volume 6 Issue 1 (January 2016), pages 86-109.
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, FPSAC 2010, San Francisco, USA.
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals, preprint, Theoretical Computer Science, Volume 479, 1 April 2013, Pages 4-21.
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

Cf. A002893, A002895, A008459, A169714.

W := proc(n,s)
    local a,ai ;
    if s = 0 then
        return 1;
    end if;
    a := 0 ;
    for ai in combinat[partition](s/2) do
        if nops(ai) <= n then
            af := [op(ai),seq(0,i=1+nops(ai)..n)] ;
            a := a+combinat[numbperm](af)*(combinat[multinomial](s/2,op(ai)))^2 ;
        end if ;
    end do;
    a ;
end proc:
A169715 := proc(n)
    W(6,2*n) ;
end proc: # R. J. Mathar, Mar 27 2012

a[n_] := SeriesCoefficient[BesselI[0, 2*Sqrt[x]]^6, {x, 0, n}]*n!^2; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 30 2013, after Peter Bala *)
max = 17; Total /@ MatrixPower[Table[Binomial[n, k]^2, {n, 0, max}, {k, 0, max}], 5] (* Jean-François Alcover, Mar 24 2015, after Peter Bala *)

Sum_{n>=0} a(n)*x^n/n!^2 = (Sum_{n>=0} x^n/n!^2)^6 = BesselI(0, 2*sqrt(x))^6. - Peter Bala, Mar 05 2013
Recurrence: n^5*a(n) = 2*(2*n-1)*(14*n^4 - 28*n^3 + 28*n^2 - 14*n + 3)*a(n-1) - 4*(n-1)^3*(196*n^2 - 392*n + 255)*a(n-2) + 1152*(n-2)^2*(n-1)^2*(2*n-3)*a(n-3). - Vaclav Kotesovec, Mar 09 2014
a(n) ~ 3^(2*n+3) * 4^(n-1) / (Pi*n)^(5/2). - Vaclav Kotesovec, Mar 09 2014

cp's OEIS Frontend

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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A287316 Square array A(n,k) = (n!)^2 [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.

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A169715 The function W_6(2n) (see Borwein et al. reference for definition).

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A287317 Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

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A169711 The function W_n(6) (see Borwein et al. reference for definition).

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