A246189 Number of endofunctions on [n] where the smallest cycle length equals 2.

1, 6, 51, 580, 8265, 141246, 2810437, 63748728, 1622579985, 45775778950, 1417347491241, 47776074289164, 1741386177576409, 68238497945688630, 2860625245955274225, 127736893134458097136, 6052712065187733972513, 303322427195785592735502, 16028016368907840953165425

Offset: 2

Author

Alois P. Heinz, Aug 18 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..200

Crossrefs

Column k=2 of A246049.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 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, k), j=0..n):
a:= n-> A(n, 2) -A(n, 3):
seq(a(n), n=2..25);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
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}]]];
A[n_, k_] := Sum[Binomial[n - 1, j - 1] n^(n - j) b[j, k], {j, 0, n}];
a[n_] := A[n, 2] - A[n, 3];
a /@ Range[2, 25] (* Jean-François Alcover, Dec 28 2020, after Alois P. Heinz *)

Formula

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