A346410 a(n) = (n!)^2 * Sum_{k=0..n-1} 1 / ((n-k) * k!)^2.

0, 1, 5, 22, 152, 2001, 45097, 1527506, 71864928, 4466430513, 353828600029, 34770661312190, 4148422395161464, 590479899466175681, 98824492409739430401, 19209838771051338898234, 4291488438323868507946880, 1091819942877526843993466529, 313819508664449992611846900981

Offset: 0

Author

Ilya Gutkovskiy, Jul 15 2021

Keywords

• nonn

Crossrefs

Cf. A002104, A006040, A066998, A193563, A336291, A346411.

Programs

Mathematica

Table[(n!)^2 Sum[1/((n - k) k!)^2, {k, 0, n - 1}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[PolyLog[2, x] BesselI[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

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