A290576 Apéry-like numbers Sum_{k=0..n} Sum_{l=0..n} (C(n,k)^2*C(n,l)*C(k,l)*C(k+l,n)).

1, 3, 27, 309, 4059, 57753, 866349, 13492251, 216077787, 3536145057, 58875891777, 994150929951, 16984143140589, 293036113226223, 5098773125244483, 89368239352074309, 1576424378494272987, 27964450505226314673, 498550055166916502121

Offset: 0

Author

Hugo Pfoertner, Aug 06 2017

Keywords

• nonn
• easy

Comments

Sequence zeta (formula 4.12) in Almkvist, Straten, Zudilin article.

Links

• Robert Israel, Table of n, a(n) for n = 0..779
• G. Almkvist, D. van Straten, and W. Zudilin, Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinburgh Math. Soc.54 (2) (2011), 273-295.
• Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See zeta p. 3.
• 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.

Crossrefs

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.

Programs

Maple

f:= gfun:-rectoproc({a(0)=1, a(1)=3, a(n+1) = ( (2*n+1)*(9*n^2+9*n+3)*a(n) + 27*n^3*a(n-1) ) / (n+1)^3}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Aug 07 2017

Mathematica

Table[Sum[Sum[(Binomial[n, k]^2*Binomial[n, j] Binomial[k, j] Binomial[k + j, n]), {j, 0, n} ], {k, 0, n}], {n, 0, 18}] (* Michael De Vlieger, Aug 07 2017 *)

PARI

C=binomial;
a(n) = sum(k=0,n, sum(l=0,n, C(n,k)^2 * C(n,l) * C(k,l) * C(k+l,n) ));

Formula

a(0) = 1, a(1) = 3,

a(n+1) = ( (2*n+1)*(9*n^2+9*n+3)*a(n) + 27*n^3*a(n-1) ) / (n+1)^3.

a(n) ~ 3^(3*n/2 + 1) * (1+sqrt(3))^(2*n+1) / (2^(n + 5/2) * (Pi*n)^(3/2)). - Vaclav Kotesovec, Jul 10 2021

G.f.: hypergeom([1/12,5/12],[1],(12*x/(1-6*x-27*x^2))^3)^2/(1-6*x-27*x^2)^(1/2). - Mark van Hoeij, Nov 11 2022