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%I #5 Jul 13 2020 07:53:26
%S 1,1,4,27,292,4425,89106,2280901,71928872,2728450017,122145511510,
%T 6354868381521,379376236939404,25710543779239501,1960001963705060926,
%U 166753195643254805565,15724259680648667902096,1633462474351643785483457,185931510605274506452763166
%N a(0) = 1; a(n) = n * Sum_{k=0..n-1} binomial(n-1,k)^2 * a(k).
%F a(n) = (n!)^2 * [x^n] exp(sqrt(x) * BesselI(1,2*sqrt(x))).
%t a[0] = 1; a[n_] := a[n] = n Sum[Binomial[n - 1, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
%t nmax = 18; CoefficientList[Series[Exp[Sqrt[x] BesselI[1, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A000248, A023998.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 12 2020