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%I #8 Sep 04 2020 10:06:54
%S 0,1,5,48,909,28836,1371384,91308708,8106024861,925225277004,
%T 132007041682380,23019553116101268,4817014157800460664,
%U 1191268407723761654964,343706793228408937835772,114423311913128119741898268,43534429651349601213257298621,18771927426013054800534345817884,9106204442628918977341144456510260
%N a(0) = 0; a(n) = 3^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 3^(k-1) * (n-k) * a(n-k).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = -log((4 - BesselI(0,2*sqrt(3*x))) / 3).
%t a[0] = 0; a[n_] := a[n] = 3^(n - 1) + (1/n) Sum[Binomial[n, k]^2 3^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n],{n, 0, 18}]
%t nmax = 18; CoefficientList[Series[-Log[(4 - BesselI[0, 2 Sqrt[3 x]])/3], {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A102223, A201355, A333981, A333983, A333984, A333985, A337593.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Sep 04 2020
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