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%I #7 Jan 27 2021 18:44:02
%S 1,3,21,271,5649,174051,7447573,422836191,30767443521,2792343036259,
%T 309252314731701,41051709426337743,6434479982900111761,
%U 1175819833620882461571,247785659825802622964469,59649892258930263778729951,16268290830606063971956320513
%N a(n) = (n!)^2 * Sum_{k=0..n} (k+1) / ((n-k)!)^2.
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - x)^2.
%t Table[n!^2 Sum[(k + 1)/(n - k)!^2, {k, 0, n}], {n, 0, 16}]
%t nmax = 16; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - x)^2, {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A001339, A006040.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jan 27 2021
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