A334886 - OEIS

%I #24 Oct 23 2022 23:33:37

%S 0,12,210,4438,104825,13276637,70543291,67890874657,766399019471,

%T 331100496521629,1938622271127289,632711558774104687,

%U 18983793845583749117,7475161875743183448469,32609977844866666501811,5014348056100175667015179,5642363887250888549594351

%N Numerator of 2*Sum_{k=0..n} binomial(n,k)^2*binomial(n+k,k)^2*(H(n+k)-H(n-k)) where H(n)=Sum_{k=1..n} 1/k.

%C Conjecture: a(p-1) == 0 (mod p^2) for all primes p >= 5 (checked up to p = 499). - _Peter Bala_, Oct 23 2022

%H Ira Gessel, <a href="https://doi.org/10.1016/0022-314X(82)90071-3">Some congruences for Apéry numbers</a>, Journal of Number Theory 14 (1982) 362-368. See b(n).

%H Eric Rowland, Reem Yassawi and Christian Krattenthaler, <a href="https://arxiv.org/abs/2005.04801">Lucas congruences for the Apéry numbers modulo p^2</a>, arXiv:2005.04801 [math.NT], 2020. See A'(n).

%p H:= proc(n) option remember; `if`(n=0, 0, 1/n+H(n-1)) end:

%p a:= n-> numer(2*add(binomial(n, k)^2*binomial(n+k, k)^2*(H(n+k)-H(n-k)), k=0..n)):

%p seq(a(n), n=0..17);  # _Alois P. Heinz_, May 14 2020

%t a[n_] := Numerator[2 * Sum[Binomial[n, k]^2 * Binomial[n + k, k]^2 * (HarmonicNumber[n + k] - HarmonicNumber[n - k]), {k, 0, n}]]; Array[a, 17, 0] (* _Amiram Eldar_, May 14 2020 *)

%o (PARI) H(n) = sum(k=1, n, 1/k);

%o a(n) = numerator(2*sum(k=0, n, binomial(n,k)^2*binomial(n+k,k)^2*(H(n+k)-H(n-k))));

%Y Cf. A001008, A002805, A334887 (denominators).

%K nonn,frac

%O 0,2

%A _Michel Marcus_, May 14 2020