A323290 Numerator of the sum of inverse products of cycle sizes in all permutations of [n].

1, 1, 3, 19, 107, 641, 51103, 1897879, 7860361, 505249081, 40865339743, 1355547261301, 244350418462637, 34907820791828741, 1949845703291363567, 1136592473036395958917, 31690844708764028510969, 2681369908698254192692979, 768531714669026186032238737

Offset: 0

Author

Alois P. Heinz, Jan 09 2019

Keywords

• nonn
• frac

Examples

			1/1, 1/1, 3/2, 19/6, 107/12, 641/20, 51103/360, 1897879/2520, 7860361/1680, 505249081/15120, 40865339743/151200, ... = A323290/A323291
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..270

Crossrefs

Denominators: A323291.

Cf. A000142, A177208, A177209, A322364, A322365, A322380, A322381, A323339, A323340.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*binomial(n-1, j-1)*(j-1)!/j, j=1..n))
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..20);

Mathematica

nmax = 20; Numerator[CoefficientList[Series[Exp[PolyLog[2, x]], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Feb 12 2024 *)

Formula

E.g.f.: exp(polylog(2,x)) (for fractions A323290(n)/A323291(n)). - Vaclav Kotesovec, Feb 12 2024

A323290(n)/A323291(n) ~ exp(Pi^2/6) * n! / n^2. - Vaclav Kotesovec, Feb 14 2024