%I #9 May 04 2024 12:00:09
%S 1,1,8,117,3184,134025,8141436,672837277,72634878016,9923765772177,
%T 1673881314096700,341631408064928421,82978986493779894288,
%U 23653894531273155603961,7819996460332550715977588,2967815528758036870644773925,1281517958938232539844046259456
%N Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / (1 - x * BesselI(0,2*sqrt(x))).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / (1 - Sum_{n>=1} x^n / ((n-1)!)^2).
%F a(0) = 1; a(n) = Sum_{k=0..n-1} (binomial(n,k) * (n-k))^2 * a(k).
%F a(n) ~ n!^2 / ((1 + r^(3/2)*BesselI(1, 2*sqrt(r))) * r^n), where r = 0.592860029867912878114616561736048937618032595935338954527835... is the root of the equation r*BesselI(0, 2*sqrt(r)) = 1. - _Vaclav Kotesovec_, May 04 2024
%t nmax = 16; CoefficientList[Series[1/(1 - x BesselI[0, 2 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!^2
%t a[0] = 1; a[n_] := a[n] = Sum[(Binomial[n, k] (n - k))^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
%o (PARI) a(n) = {n!^2*polcoef(1 / (1 - sum(k=1, n, x^k / ((k-1)!)^2) + O(x*x^n)), n)} \\ _Andrew Howroyd_, Mar 04 2021
%Y Cf. A006153, A101514, A336228, A337591.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Mar 04 2021