A246192 - OEIS

%I #9 Dec 28 2020 09:52:57

%S 24,720,17640,430080,11022480,302472576,8937981360,284552040960,

%T 9743091569640,357820740076800,14051646110285784,588177615908413440,

%U 26161789829441054880,1232890909824506204160,61387038018996808785120,3221070809733138102829056

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

%H Alois P. Heinz, <a href="/A246192/b246192.txt">Table of n, a(n) for n = 5..200</a>

%F a(n) ~ (exp(-25/12) - exp(-137/60)) * n^n. - _Vaclav Kotesovec_, Aug 21 2014

%p with(combinat):

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,

%p       add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*

%p       b(n-i*j, i+1), j=0..n/i)))

%p     end:

%p A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):

%p a:= n-> A(n, 5) -A(n, 6):

%p seq(a(n), n=5..25);

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t 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 A[n_, k_] :=

%t  Sum[Binomial[n - 1, j - 1] n^(n - j) b[j, k], {j, 0, n}];

%t a[n_] := A[n, 5] - A[n, 6];

%t a /@ Range[5, 25] (* _Jean-François Alcover_, Dec 28 2020, after _Alois P. Heinz_ *)

%Y Column k=5 of A246049.

%K nonn

%O 5,1

%A _Alois P. Heinz_, Aug 18 2014