A336809 a(n) = (n!)^2 * Sum_{k=0..n} (k+1) / ((n-k)!)^2.

1, 3, 21, 271, 5649, 174051, 7447573, 422836191, 30767443521, 2792343036259, 309252314731701, 41051709426337743, 6434479982900111761, 1175819833620882461571, 247785659825802622964469, 59649892258930263778729951, 16268290830606063971956320513

Offset: 0

Author

Ilya Gutkovskiy, Jan 27 2021

Keywords

• nonn

Crossrefs

Cf. A001339, A006040.

Programs

Mathematica

Table[n!^2 Sum[(k + 1)/(n - k)!^2, {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - x)^2, {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - x)^2.

a(n) ~ BesselI(0,2) * n!^2 * n. - Vaclav Kotesovec, Jul 11 2025