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%I #4 Jan 25 2021 19:05:01
%S 1,1,8,124,3456,150656,9453056,807373568,90066059264,12716049596416,
%T 2216452086693888,467465806422867968,117332539562036035584,
%U 34562989958399757647872,11807922834511544081973248,4630865359842075866336067584,2066370767828213666946077425664
%N a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 4^(n-k-1) * a(k).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = 4 / (5 - BesselI(0,4*sqrt(x))).
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 4^(n - k - 1) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
%t nmax = 16; CoefficientList[Series[4/(5 - BesselI[0, 4 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A102221, A326324, A340886, A340887.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 25 2021