%I #20 Aug 26 2021 11:18:46
%S 1,1,1,1,1,5,1,35,14,210,42,462,462,6006,858,4290,156,145860,29172,
%T 2771340,32604,554268,554268,12748164,12748164,318704100,63740820,
%U 4574700,191222460,1459329300,326203020,34381798308,343817983080,68763596616,343817983080,12033629407800
%N Denominator 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.
%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-> denom(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..37); # _Alois P. Heinz_, May 14 2020
%t a[n_] := Denominator[2 * Sum[Binomial[n, k]^2 * Binomial[n + k, k]^2 * (HarmonicNumber[n + k] - HarmonicNumber[n - k]), {k, 0, n}]]; Array[a, 36, 0] (* _Amiram Eldar_, May 14 2020 *)
%o (PARI) H(n) = sum(k=1, n, 1/k);
%o a(n) = denominator(2*sum(k=0, n, binomial(n,k)^2*binomial(n+k,k)^2*(H(n+k)-H(n-k))));
%Y Cf. A001008, A002805, A334886 (numerators).
%K nonn,frac
%O 0,6
%A _Michel Marcus_, May 14 2020