A246050 Number of endofunctions on [2n] where the smallest cycle length equals n.

1, 3, 51, 4360, 861420, 302472576, 165549605760, 130241382036480, 139296260790086400, 194427690066299289600, 343266609438110040883200, 747889929370001008617062400, 1971026055567996899374212710400, 6180432763819774878006029844480000

Offset: 0

Author

Alois P. Heinz, Aug 11 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..190

Crossrefs

Cf. A241982, A246049, A243098.

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-> `if`(n=0, 1, A(2*n, n) -A(2*n, n+1)):
seq(a(n), n=0..15);

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}, Array[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_] := If[n == 0, 1, A[2*n, n] - A[2*n, n+1]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

Formula

a(n) = A246049(2n,n) = A243098(2n,n).

a(n) ~ 2^(3*n-1/2) * n^(2*n-1) / exp(n). - Vaclav Kotesovec, Aug 19 2014