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%I #4 Sep 02 2020 19:24:20
%S 1,1,6,51,760,15545,428256,15043483,653049664,34204348305,
%T 2118834917200,152834879685851,12670536337934256,1194143629239156505,
%U 126753440317516749660,15031687739886065433375,1977667235694725269563136,286890421090357737699794209,45637300134026406622214264592
%N a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k^3 * a(n-k).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x * BesselI(0,2*sqrt(x))).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} n^2 * x^n / (n!)^2).
%t a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%t nmax = 18; CoefficientList[Series[Exp[x BesselI[0, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A033462, A336227.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Sep 02 2020
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