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A291111 Number of endofunctions on [n] such that the LCM of their cycle lengths equals five.

%I A291111 #9 Dec 29 2020 09:04:26
%S A291111 0,0,0,0,0,24,864,24192,653184,18144000,531365184,16563076992,
%T A291111 551172885120,19580825392128,741547690884000,29873618711000064,
%U A291111 1277121733631347968,57795924098354577408,2762004604309125452928,139058300756829929472000,7359536118308288021017344
%N A291111 Number of endofunctions on [n] such that the LCM of their cycle lengths equals five.
%H A291111 Alois P. Heinz, <a href="/A291111/b291111.txt">Table of n, a(n) for n = 0..387</a>
%F A291111 a(n) ~ (2*exp(6/5)-exp(1)) * n^(n-1). - _Vaclav Kotesovec_, Aug 18 2017
%p A291111 b:= proc(n, m) option remember; (k-> `if`(m>k, 0,
%p A291111       `if`(n=0, `if`(m=k, 1, 0), add(b(n-j, ilcm(m, j))
%p A291111        *binomial(n-1, j-1)*(j-1)!, j=1..n))))(5)
%p A291111     end:
%p A291111 a:= n-> add(b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A291111 seq(a(n), n=0..22);
%t A291111 b[n_, m_] := b[n, m] = With[{k = 5}, If[m > k, 0, If[n == 0, If[m == k, 1, 0], Sum[b[n-j, LCM[m, j]] Binomial[n-1, j-1] (j-1)!, {j, 1, n}]]]];
%t A291111 a[n_] := If[n == 0, 0, Sum[b[j, 1] n^(n-j) Binomial[n-1, j-1], {j, 0, n}]];
%t A291111 a /@ Range[0, 22] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)
%Y A291111 Column k=5 of A222029.
%K A291111 nonn
%O A291111 0,6
%A A291111 _Alois P. Heinz_, Aug 17 2017