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%I #8 Jan 27 2021 18:43:47
%S 1,5,81,2917,186689,18668901,2688321745,526911062021,134889231877377,
%T 43704111128270149,17481644451308059601,8461115914433100846885,
%U 4873602766713466087805761,3294555470298303075356694437,2582931488713869611079648438609,2324638339842482649971683594748101
%N a(n) = (n!)^2 * Sum_{k=0..n} 4^(n-k) / (k!)^2.
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - 4*x).
%F a(0) = 1; a(n) = 4 * n^2 * a(n-1) + 1.
%t Table[n!^2 Sum[4^(n - k)/k!^2, {k, 0, n}], {n, 0, 15}]
%t nmax = 15; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - 4 x), {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A006040, A056545, A336804, A336805, A336808.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jan 27 2021
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