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A290576
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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)).
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46
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1, 3, 27, 309, 4059, 57753, 866349, 13492251, 216077787, 3536145057, 58875891777, 994150929951, 16984143140589, 293036113226223, 5098773125244483, 89368239352074309, 1576424378494272987, 27964450505226314673, 498550055166916502121
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OFFSET
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0,2
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COMMENTS
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Sequence zeta (formula 4.12) in Almkvist, Straten, Zudilin article.
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LINKS
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Robert Israel, Table of n, a(n) for n = 0..779
G. Almkvist, D. van Straten, W. Zudilin, Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinburgh Math. Soc.54 (2) (2011), 273-295.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
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FORMULA
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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) ));
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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Hugo Pfoertner, Aug 06 2017
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STATUS
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approved
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