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A346066 Sum of GCD of cycle lengths over all permutations of [n].

%I A346066 #16 Mar 06 2022 04:44:10
%S A346066 0,1,3,10,45,216,1505,9360,84105,730240,7715169,76204800,1090114025,
%T A346066 11975040000,185501455425,2791872219136,45361870178625,
%U A346066 690452066304000,14415096609538625,236887827111936000,5448878874163974249,108418310412206080000,2381309423564793710625
%N A346066 Sum of GCD of cycle lengths over all permutations of [n].
%H A346066 Alois P. Heinz, <a href="/A346066/b346066.txt">Table of n, a(n) for n = 0..450</a>
%H A346066 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A346066 a(n) = Sum_{k=1..n} k * A346085(n,k).
%e A346066 a(3) = 10 = 3+3+1+1+1+1: (123), (132), (1)(23), (13)(2), (12)(3), (1)(2)(3).
%p A346066 b:= proc(n, g) option remember; `if`(n=0, g, add((j-1)!
%p A346066       *b(n-j, igcd(g, j))*binomial(n-1, j-1), j=1..n))
%p A346066     end:
%p A346066 a:= n-> b(n, 0):
%p A346066 seq(a(n), n=0..24);
%t A346066 b[n_, g_] := b[n, g] = If[n == 0, g, Sum[(j - 1)!*
%t A346066      b[n - j, GCD[g, j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
%t A346066 a[n_] := b[n, 0];
%t A346066 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Mar 06 2022, after _Alois P. Heinz_ *)
%Y A346066 Cf. A060014 (the same for LCM), A346085.
%K A346066 nonn
%O A346066 0,3
%A A346066 _Alois P. Heinz_, Jul 03 2021