This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A336809 #9 Jul 11 2025 06:15:18
%S A336809 1,3,21,271,5649,174051,7447573,422836191,30767443521,2792343036259,
%T A336809 309252314731701,41051709426337743,6434479982900111761,
%U A336809 1175819833620882461571,247785659825802622964469,59649892258930263778729951,16268290830606063971956320513
%N A336809 a(n) = (n!)^2 * Sum_{k=0..n} (k+1) / ((n-k)!)^2.
%F A336809 Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - x)^2.
%F A336809 a(n) ~ BesselI(0,2) * n!^2 * n. - _Vaclav Kotesovec_, Jul 11 2025
%t A336809 Table[n!^2 Sum[(k + 1)/(n - k)!^2, {k, 0, n}], {n, 0, 16}]
%t A336809 nmax = 16; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - x)^2, {x, 0, nmax}], x] Range[0, nmax]!^2
%Y A336809 Cf. A001339, A006040.
%K A336809 nonn
%O A336809 0,2
%A A336809 _Ilya Gutkovskiy_, Jan 27 2021