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%I #8 Jan 27 2021 18:43:35
%S 1,3,25,451,14433,721651,51958873,5091969555,651772103041,
%T 105587080692643,21117416138528601,5110414705523921443,
%U 1471799435190889375585,497468209094520608947731,195007537965052078707510553,87753392084273435418379748851,44929736747147998934210431411713
%N a(n) = (n!)^2 * Sum_{k=0..n} 2^(n-k) / (k!)^2.
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - 2*x).
%F a(0) = 1; a(n) = 2 * n^2 * a(n-1) + 1.
%t Table[n!^2 Sum[2^(n - k)/k!^2, {k, 0, n}], {n, 0, 16}]
%t nmax = 16; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A006040, A010844, A228513, A336805, A336807, A336808.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jan 27 2021
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