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A246192 Number of endofunctions on [n] where the smallest cycle length equals 5.

%I A246192 #9 Dec 28 2020 09:52:57
%S A246192 24,720,17640,430080,11022480,302472576,8937981360,284552040960,
%T A246192 9743091569640,357820740076800,14051646110285784,588177615908413440,
%U A246192 26161789829441054880,1232890909824506204160,61387038018996808785120,3221070809733138102829056
%N A246192 Number of endofunctions on [n] where the smallest cycle length equals 5.
%H A246192 Alois P. Heinz, <a href="/A246192/b246192.txt">Table of n, a(n) for n = 5..200</a>
%F A246192 a(n) ~ (exp(-25/12) - exp(-137/60)) * n^n. - _Vaclav Kotesovec_, Aug 21 2014
%p A246192 with(combinat):
%p A246192 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
%p A246192       add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p A246192       b(n-i*j, i+1), j=0..n/i)))
%p A246192     end:
%p A246192 A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):
%p A246192 a:= n-> A(n, 5) -A(n, 6):
%p A246192 seq(a(n), n=5..25);
%t A246192 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A246192 b[n_, i_] := b[n, i] = If[n==0, 1, If[i>n, 0, Sum[(i - 1)!^j multinomial[n, Join[{n - i*j}, Table[i, {j}]]]/j! b[n - i*j, i + 1], {j, 0, n/i}]]];
%t A246192 A[n_, k_] :=
%t A246192  Sum[Binomial[n - 1, j - 1] n^(n - j) b[j, k], {j, 0, n}];
%t A246192 a[n_] := A[n, 5] - A[n, 6];
%t A246192 a /@ Range[5, 25] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)
%Y A246192 Column k=5 of A246049.
%K A246192 nonn
%O A246192 5,1
%A A246192 _Alois P. Heinz_, Aug 18 2014