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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)).

%I A290576 #52 May 04 2023 04:37:01
%S A290576 1,3,27,309,4059,57753,866349,13492251,216077787,3536145057,
%T A290576 58875891777,994150929951,16984143140589,293036113226223,
%U A290576 5098773125244483,89368239352074309,1576424378494272987,27964450505226314673,498550055166916502121
%N 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)).
%C A290576 Sequence zeta (formula 4.12) in Almkvist, Straten, Zudilin article.
%H A290576 Robert Israel, <a href="/A290576/b290576.txt">Table of n, a(n) for n = 0..779</a>
%H A290576 G. Almkvist, D. van Straten, and W. Zudilin, <a href="https://doi.org/10.1017/S0013091509000959">Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations</a>, Proc. Edinburgh Math. Soc.54 (2) (2011), 273-295.
%H A290576 Ofir Gorodetsky, <a href="https://arxiv.org/abs/2102.11839">New representations for all sporadic Apéry-like sequences, with applications to congruences</a>, arXiv:2102.11839 [math.NT], 2021. See zeta p. 3.
%H A290576 Timothy Huber, Daniel Schultz, and Dongxi Ye, <a href="https://doi.org/10.4064/aa220621-19-12">Ramanujan-Sato series for 1/pi</a>, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
%H A290576 Amita Malik and Armin Straub, <a href="https://doi.org/10.1007/s40993-016-0036-8">Divisibility properties of sporadic Apéry-like numbers</a>, Research in Number Theory, 2016, 2:5.
%F A290576 a(0) = 1, a(1) = 3,
%F A290576 a(n+1) = ( (2*n+1)*(9*n^2+9*n+3)*a(n) + 27*n^3*a(n-1) ) / (n+1)^3.
%F A290576 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
%F A290576 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
%p A290576 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):
%p A290576 map(f, [$0..30]); # _Robert Israel_, Aug 07 2017
%t A290576 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 *)
%o A290576 (PARI) C=binomial;
%o A290576 a(n) = sum(k=0,n, sum(l=0,n, C(n,k)^2 * C(n,l) * C(k,l) * C(k+l,n) ));
%Y A290576 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.)
%Y A290576 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.
%K A290576 nonn,easy
%O A290576 0,2
%A A290576 _Hugo Pfoertner_, Aug 06 2017