A246215 Number of endofunctions on [n] where the largest cycle length equals 5.

24, 864, 24696, 688128, 19840464, 604727424, 19632956112, 680195957760, 25130679950376, 988325574652416, 41277744231187464, 1826323584590389248, 85391029667937905184, 4209030460729215184896, 218223423136426488339744, 11875233973816788160610304

Offset: 5

Author

Alois P. Heinz, Aug 19 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..200

Crossrefs

Column k=5 of A241981.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1), j=0..n/i)))
    end:
A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, min(j, k)), j=0..n):
a:= n-> A(n, 5) -A(n, 4):
seq(a(n), n=5..25);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 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}]]];
A[n_, k_] := Sum[Binomial[n-1, j-1] n^(n-j) b[j, Min[j, k]], {j, 0, n}];
a[n_] := A[n, 5] - A[n, 4];
a /@ Range[5, 25] (* Jean-François Alcover, Dec 28 2020, after Alois P. Heinz *)

Formula

a(n) ~ (5*exp(137/60) - 4*exp(25/12)) * n^(n-1). - Vaclav Kotesovec, Aug 21 2014