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

3,1

Links

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

Formula

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

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 *)

Crossrefs

Column k=3 of A241981.

Sequence in context: A230131 A115418 A163952 * A022028 A013776 A174491

Adjacent sequences: A246210 A246211 A246212 * A246214 A246215 A246216

Keyword

nonn

Author

Alois P. Heinz, Aug 19 2014

Status

approved