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A374262 Number of permutations of [n] such that the number of cycles of length k is a multiple of k for every k.

%I A374262 #20 Jul 04 2024 18:34:04
%S A374262 1,1,1,1,4,16,46,106,316,3564,27756,141516,556656,6678816,73015944,
%T A374262 521124696,6144018336,75200767776,677927254176,4642387894944,
%U A374262 75217104395136,1167068528384256,12348761954020416,97377968145352896,882819252604721664,66882151986021043200
%N A374262 Number of permutations of [n] such that the number of cycles of length k is a multiple of k for every k.
%H A374262 Alois P. Heinz, <a href="/A374262/b374262.txt">Table of n, a(n) for n = 0..470</a>
%H A374262 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A374262 a(4) = 4: (1)(2)(3)(4), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3).
%p A374262 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A374262       add(combinat[multinomial](n, i$i*j, n-i^2*j)*
%p A374262       b(n-i^2*j, i-1)*(i-1)!^(i*j)/(i*j)!, j=0..n/i^2))
%p A374262     end:
%p A374262 a:= n-> b(n$2):
%p A374262 seq(a(n), n=0..25);
%Y A374262 Cf. A000704, A130268, A372545, A372579, A374292, A374320.
%K A374262 nonn
%O A374262 0,5
%A A374262 _Alois P. Heinz_, Jul 01 2024