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A382780 Sum of the orders of all permutations of [n] with distinct cycle lengths.

%I A382780 #13 May 13 2025 11:21:24
%S A382780 1,1,2,12,48,360,2520,22680,221760,2298240,28425600,385862400,
%T A382780 5269017600,80951270400,1347631084800,21565729785600,413922526617600,
%U A382780 8409043612569600,172028224598630400,3765253760710041600,84080417596471296000,1935910813364656128000
%N A382780 Sum of the orders of all permutations of [n] with distinct cycle lengths.
%H A382780 Alois P. Heinz, <a href="/A382780/b382780.txt">Table of n, a(n) for n = 0..170</a>
%H A382780 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A382780 a(3) = 12 = 2+2+2+3+3: (1)(23), (13)(2), (12)(3), (123), (132).
%p A382780 b:= proc(n, i, m) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, m,
%p A382780       b(n, i-1, m)+b(n-i, min(i-1, n-i), ilcm(i,m))*binomial(n, i)*(i-1)!))
%p A382780     end:
%p A382780 a:= n-> b(n$2, 1):
%p A382780 seq(a(n), n=0..22);
%Y A382780 Cf. A000142, A000793, A007838, A060014, A382781.
%K A382780 nonn
%O A382780 0,3
%A A382780 _Alois P. Heinz_, May 11 2025