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 A336639 #12 Jul 11 2025 04:25:10
%S A336639 1,4,36,544,12196,377904,15438816,803602944,51908768676,4074743122384,
%T A336639 382079412133936,42184889139337344,5417567866536188896,
%U A336639 800808722921088352384,135006904500993157933056,25751088570167886107910144,5517695042810314282550432676
%N A336639 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^4.
%C A336639 In general, if k>=1 and Sum_{n>=0} a(n) * x^n / n!^2 = 1 / BesselJ(0, 2*sqrt(x))^k, then a(n) ~ n!^2 * n^(k-1) / ((k-1)! * r^(n + k/2) * BesselJ(1, 2*sqrt(r))^k), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4. - _Vaclav Kotesovec_, Jul 11 2025
%F A336639 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * A002895(k) * a(n-k).
%F A336639 a(n) ~ n!^2 * n^3 / (6 * r^(n+2) * BesselJ(1, 2*sqrt(r))^4), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - _Vaclav Kotesovec_, Jul 11 2025
%t A336639 nmax = 16; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^4, {x, 0, nmax}], x] Range[0, nmax]!^2
%t A336639 a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 Binomial[2 k, k] HypergeometricPFQ[{1/2, -k, -k, -k}, {1, 1, 1/2 - k}, 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]
%Y A336639 Cf. A000275, A002895, A336271, A336638.
%Y A336639 Column k=4 of A340986.
%K A336639 nonn
%O A336639 0,2
%A A336639 _Ilya Gutkovskiy_, Jul 28 2020