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A374292 Number of permutations of [n] such that the number of cycles of length k is zero or a divisor of k for every k.

%I A374292 #12 Jul 04 2024 17:45:24
%S A374292 1,1,1,5,17,89,474,3324,28440,253448,2476700,26876420,328110540,
%T A374292 4207321260,58468831680,877439227560,14214209548560,239870470655760,
%U A374292 4285924637475600,81381169697904720,1636049164466934000,34301061146870607600,750389221227585139200
%N A374292 Number of permutations of [n] such that the number of cycles of length k is zero or a divisor of k for every k.
%H A374292 Alois P. Heinz, <a href="/A374292/b374292.txt">Table of n, a(n) for n = 0..450</a>
%H A374292 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A374292 a(3) = 5: (1)(2,3), (1,2)(3), (1,3)(2), (1,2,3), (1,3,2).
%p A374292 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A374292       add(`if`(j=0 or irem(i, j)=0, b(n-i*j, i-1)*(i-1)!^j/j!*
%p A374292       combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
%p A374292     end:
%p A374292 a:= n-> b(n$2):
%p A374292 seq(a(n), n=0..25);
%Y A374292 Cf. A372545, A372579, A374262, A374319.
%K A374292 nonn
%O A374292 0,4
%A A374292 _Alois P. Heinz_, Jul 02 2024