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%I #5 Sep 02 2020 19:24:38
%S 1,1,6,58,920,21176,654960,26114768,1298070912,78359732608,
%T 5630565514496,473796572027648,46060380961356800,5114737212582603776,
%U 642502387594286036992,90542358999393528670208,14209873001490130067095552,2467784343879850163370295296,471558856613839054976849608704
%N a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 4^(k-1) * a(n-k).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,4*sqrt(x)) - 1) / 4).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 4^(n-1) * x^n / (n!)^2).
%t a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%t nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 4 Sqrt[x]] - 1)/4], {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A004213, A337592, A337593, A337595, A337597.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Sep 02 2020