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%I #8 Jan 27 2021 18:43:40
%S 1,4,49,1324,63553,4766476,514779409,75672573124,14529134039809,
%T 3530579571673588,1059173871502076401,384480115355253733564,
%U 166095409833469612899649,84210372785569093740122044,49515699197914627119191761873,33423096958592373305454439264276,25668938464198942698589009354963969
%N a(n) = (n!)^2 * Sum_{k=0..n} 3^(n-k) / (k!)^2.
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - 3*x).
%F a(0) = 1; a(n) = 3 * n^2 * a(n-1) + 1.
%t Table[n!^2 Sum[3^(n - k)/k!^2, {k, 0, n}], {n, 0, 16}]
%t nmax = 16; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - 3 x), {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A006040, A010845, A336804, A336807, A336808.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jan 27 2021