This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A336638 #9 Jul 11 2025 04:24:44
%S A336638 1,3,21,255,4725,123903,4368729,199467243,11455187445,808475761695,
%T A336638 68805857523321,6950458374996843,822292004658568761,
%U A336638 112639503374757412875,17688916392275574761805,3157133540377493872350855,635546443798928578953138165
%N A336638 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^3.
%F A336638 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * A002893(k) * a(n-k).
%F A336638 a(n) ~ n!^2 * n^2 / (2 * r^(n + 3/2) * BesselJ(1, 2*sqrt(r))^3), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - _Vaclav Kotesovec_, Jul 11 2025
%t A336638 nmax = 16; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^3, {x, 0, nmax}], x] Range[0, nmax]!^2
%t A336638 a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 HypergeometricPFQ[{1/2, -k, -k}, {1, 1}, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]
%Y A336638 Cf. A000275, A002893, A336271, A336639.
%Y A336638 Column k=3 of A340986.
%K A336638 nonn
%O A336638 0,2
%A A336638 _Ilya Gutkovskiy_, Jul 28 2020