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

%I A323290 #15 Feb 14 2024 05:41:51
%S A323290 1,1,3,19,107,641,51103,1897879,7860361,505249081,40865339743,
%T A323290 1355547261301,244350418462637,34907820791828741,1949845703291363567,
%U A323290 1136592473036395958917,31690844708764028510969,2681369908698254192692979,768531714669026186032238737
%N A323290 Numerator of the sum of inverse products of cycle sizes in all permutations of [n].
%H A323290 Alois P. Heinz, <a href="/A323290/b323290.txt">Table of n, a(n) for n = 0..270</a>
%F A323290 E.g.f.: exp(polylog(2,x)) (for fractions A323290(n)/A323291(n)). - _Vaclav Kotesovec_, Feb 12 2024
%F A323290 A323290(n)/A323291(n) ~ exp(Pi^2/6) * n! / n^2. - _Vaclav Kotesovec_, Feb 14 2024
%e A323290 1/1, 1/1, 3/2, 19/6, 107/12, 641/20, 51103/360, 1897879/2520, 7860361/1680, 505249081/15120, 40865339743/151200, ... = A323290/A323291
%p A323290 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A323290       b(n-j)*binomial(n-1, j-1)*(j-1)!/j, j=1..n))
%p A323290     end:
%p A323290 a:= n-> numer(b(n)):
%p A323290 seq(a(n), n=0..20);
%t A323290 nmax = 20; Numerator[CoefficientList[Series[Exp[PolyLog[2, x]], {x, 0, nmax}], x] * Range[0, nmax]!] (* _Vaclav Kotesovec_, Feb 12 2024 *)
%Y A323290 Denominators: A323291.
%Y A323290 Cf. A000142, A177208, A177209, A322364, A322365, A322380, A322381, A323339, A323340.
%K A323290 nonn,frac
%O A323290 0,3
%A A323290 _Alois P. Heinz_, Jan 09 2019