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%I #52 May 04 2023 04:37:01
%S 1,3,27,309,4059,57753,866349,13492251,216077787,3536145057,
%T 58875891777,994150929951,16984143140589,293036113226223,
%U 5098773125244483,89368239352074309,1576424378494272987,27964450505226314673,498550055166916502121
%N 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 Sequence zeta (formula 4.12) in Almkvist, Straten, Zudilin article.
%H Robert Israel, <a href="/A290576/b290576.txt">Table of n, a(n) for n = 0..779</a>
%H 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 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 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 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 a(0) = 1, a(1) = 3,
%F a(n+1) = ( (2*n+1)*(9*n^2+9*n+3)*a(n) + 27*n^3*a(n-1) ) / (n+1)^3.
%F 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 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 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 map(f, [$0..30]); # _Robert Israel_, Aug 07 2017
%t 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 (PARI) C=binomial;
%o 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 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 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 nonn,easy
%O 0,2
%A _Hugo Pfoertner_, Aug 06 2017
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