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

2,2

Links

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

Formula

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

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 *)

Crossrefs

Column k=2 of A246049.

Sequence in context: A113352 A063169 A346667 * A293128 A304185 A215003

Adjacent sequences: A246186 A246187 A246188 * A246190 A246191 A246192

Keyword

nonn

Author

Alois P. Heinz, Aug 18 2014

Status

approved