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A382781 Sum of GCD of cycle lengths over all permutations of [n] with distinct cycle lengths.

%I A382781 #12 May 13 2025 11:17:44
%S A382781 0,1,2,9,32,170,1164,7434,62880,582336,5875200,60041520,841501440,
%T A382781 9440926560,141618778560,2222190784800,34862691548160,543348318159360,
%U A382781 11173101312844800,186494289764106240,4219768887634944000,86094733814301542400,1834643656963469721600
%N A382781 Sum of GCD of cycle lengths over all permutations of [n] with distinct cycle lengths.
%H A382781 Alois P. Heinz, <a href="/A382781/b382781.txt">Table of n, a(n) for n = 0..450</a>
%H A382781 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A382781 a(3) = 9 = 3+3+1+1+1: (123), (132), (1)(23), (13)(2), (12)(3).
%p A382781 b:= proc(n, i, m) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, m,
%p A382781       b(n, i-1, m)+b(n-i, min(i-1, n-i), igcd(i,m))*binomial(n, i)*(i-1)!))
%p A382781     end:
%p A382781 a:= n-> b(n$2, 0):
%p A382781 seq(a(n), n=0..22);
%Y A382781 Cf. A000142, A007838, A346066, A382780.
%K A382781 nonn
%O A382781 0,3
%A A382781 _Alois P. Heinz_, May 11 2025