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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)). 46


%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, 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 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.

%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 Other Apéry-like sequences are A000172, A002893, A002895, A005258, A005259, A005260, A006077, A081085, A093388, A125143, A183204, A219692, A229111, A290575.

%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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Last modified November 19 22:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)