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A357511
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a(n) = numerator of Sum_{k = 1..n} (1/k) * binomial(n,k)^2 * binomial(n+k,k)^2 for n >= 1 with a(0) = 0
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6
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0, 4, 54, 2182, 36625, 3591137, 25952409, 4220121443, 206216140401, 47128096330129, 1233722785504429, 364131107601152519, 9971452750252847789, 3611140187389794708497, 102077670374035974509597, 2922063451137950165057717, 169140610796591477659644439
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OFFSET
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0,2
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LINKS
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Table of n, a(n) for n=0..16.
A. Straub, Multivariate Apéry numbers and supercongruences of rational functions, arXiv:1401.0854 [math.NT] (2014).
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FORMULA
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Conjecture: a(p-1) == 0 (mod p^4) for all primes p >= 7 (checked up to p = 499).
Note: the Apery numbers A(n) = A005259(n) = Sum_{k = 0..n} binomial(n,k)^2 * binomial(n+k,k)^2 satisfy the supercongruence A(p-1) == 1 (mod p^3) for all primes p >= 5 (see, for example, Straub, Introduction).
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EXAMPLE
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a(13 - 1) = 9971452750252847789 = (13^4)*37*2477*24197*157433 == 0 (mod 13^4).
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MAPLE
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seq(numer(add( (1/k) * binomial(n, k)^2 * binomial(n+k, k)^2, k = 1..n )), n = 0..20);
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PROG
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(PARI) a(n) = if (n, numerator(sum(k=1, n, binomial(n, k)^2*binomial(n+k, k)^2/k)), 0); \\ Michel Marcus, Oct 04 2022
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CROSSREFS
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Cf. A005259, A357506, A357507, A357510, A357512, A357513.
Sequence in context: A182264 A355128 A355126 * A094154 A168510 A324235
Adjacent sequences: A357508 A357509 A357510 * A357512 A357513 A357514
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KEYWORD
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nonn,easy
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AUTHOR
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Peter Bala, Oct 01 2022
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STATUS
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approved
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