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%I #16 Jul 04 2024 20:15:59
%S 1,1,2,6,24,120,665,4655,37660,345660,3373629,37109919,443171498,
%T 5761229474,79709485141,1199252731963,19237203662248,327101074802216,
%U 5848216651372953,111064609625430747,2222478622302320382,46709011248199791062,1022898268873467547769
%N Number of permutations of [n] such that the number of cycles of length k is a multiple or a divisor of k for every k.
%H Alois P. Heinz, <a href="/A372545/b372545.txt">Table of n, a(n) for n = 0..450</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e a(6) = 665 = 720 - 55 counts all permutations of [6] with the exception of 15 permutations of type (12)(34)(56) and 40 permutations of type (123)(456).
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, 1, add(`if`(
%p irem(j, i)=0 or irem(i, j)=0, b(n-i*j, i-1)*(i-1)!^j/j!
%p *combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..25);
%Y Cf. A372579, A374262, A374292, A374329.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jul 04 2024
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