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A337597 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 6^(k-1) * a(n-k).

%I A337597 #8 Sep 07 2020 06:31:34
%S A337597 1,1,8,96,1896,55416,2182752,111162528,7088997888,550749341952,
%T A337597 51058009732608,5556160183592448,699989463219105792,
%U A337597 100917906076208203776,16486415052067886690304,3026039346413717945757696,619431153899977856767131648,140491838894751995366936641536,35102748598142373142198776889344
%N A337597 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 6^(k-1) * a(n-k).
%H A337597 Robert Israel, <a href="/A337597/b337597.txt">Table of n, a(n) for n = 0..200</a>
%F A337597 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,2*sqrt(6*x)) - 1) / 6).
%F A337597 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 6^(n-1) * x^n / (n!)^2).
%p A337597 S:= series(exp((BesselI(0,2*sqrt(6*x))-1)/6),x,51):
%p A337597 seq(coeff(S,x,j)*(j!)^2, j=0..50); # _Robert Israel_, Sep 06 2020
%t A337597 a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 6^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%t A337597 nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[6 x]] - 1)/6], {x, 0, nmax}], x] Range[0, nmax]!^2
%Y A337597 Cf. A005012, A337592, A337593, A337594, A337595.
%K A337597 nonn
%O A337597 0,3
%A A337597 _Ilya Gutkovskiy_, Sep 02 2020