A246213 Number of endofunctions on [n] where the largest cycle length equals 3.

2, 32, 500, 8600, 165690, 3568768, 85372280, 2251589600, 65007768650, 2041482333440, 69330316507452, 2533173484572640, 99124829660524850, 4137148176815360000, 183498069976613613680, 8620747043700633797888, 427712115490907106172050, 22350263436559575406220800

Offset: 3

Author

Alois P. Heinz, Aug 19 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..200

Crossrefs

Column k=3 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, 3) -A(n, 2):
seq(a(n), n=3..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, 3] - A[n, 2];
a /@ Range[3, 25] (* Jean-François Alcover, Dec 28 2020, after Alois P. Heinz *)

Formula

a(n) ~ (3*exp(11/6)-2*exp(3/2)) * n^(n-1). - Vaclav Kotesovec, Aug 21 2014