A357511 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.
0, 4, 54, 2182, 36625, 3591137, 25952409, 4220121443, 206216140401, 47128096330129, 1233722785504429, 364131107601152519, 9971452750252847789, 3611140187389794708497, 102077670374035974509597, 2922063451137950165057717, 169140610796591477659644439
Offset: 0
Examples
a(13 - 1) = 9971452750252847789 = (13^4)*37*2477*24197*157433 == 0 (mod 13^4).
Links
- A. Straub, Multivariate Apéry numbers and supercongruences of rational functions, arXiv:1401.0854 [math.NT] (2014).
Programs
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Maple
seq(numer(add( (1/k) * binomial(n,k)^2 * binomial(n+k,k)^2, k = 1..n )), n = 0..20);
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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
Formula
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).