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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
1, 3, 27, 309, 4059, 57753, 866349, 13492251, 216077787, 3536145057, 58875891777, 994150929951, 16984143140589, 293036113226223, 5098773125244483, 89368239352074309, 1576424378494272987, 27964450505226314673, 498550055166916502121 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

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

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 *)

PROG

(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) ));

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.)

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

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.

Sequence in context: A204821 A200903 A318108 * A291315 A078532 A264684

Adjacent sequences:  A290573 A290574 A290575 * A290577 A290578 A290579

KEYWORD

nonn,easy

AUTHOR

Hugo Pfoertner, Aug 06 2017

STATUS

approved

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Last modified September 24 16:41 EDT 2022. Contains 356943 sequences. (Running on oeis4.)