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 A336217 #6 Jul 17 2020 09:55:39
%S A336217 1,2,18,362,12946,723402,58208490,6375093258,911949196434,
%T A336217 165104835435146,36903191037412618,9980525774650881738,
%U A336217 3212329170232153022314,1213419234370490738427722,531582989226188067128503722,267336170027296964096123899962
%N A336217 a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n,k)^2 * a(k).
%F A336217 a(n) = (n!)^2 * [x^n] 1 / (1 - 2 * Sum_{k>=1} x^k / (k!)^2).
%F A336217 a(n) = (n!)^2 * [x^n] 1 / (3 - 2 * BesselI(0,2*sqrt(x))).
%F A336217 a(n) ~ (n!)^2 / (2 * BesselI(1, 2*sqrt(r)) * r^(n + 1/2)), where r = 0.4473998881770456142157108538567782213913712561... is the root of the equation 2*BesselI(0, 2*sqrt(r)) = 3. - _Vaclav Kotesovec_, Jul 17 2020
%t A336217 a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
%t A336217 nmax = 15; CoefficientList[Series[1/(1 - 2 Sum[x^k/(k!)^2, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2
%Y A336217 Cf. A004123, A102221.
%K A336217 nonn
%O A336217 0,2
%A A336217 _Ilya Gutkovskiy_, Jul 12 2020