A241982 Number of endofunctions on [2n] where the largest cycle length equals n.

1, 3, 93, 8600, 1719060, 604727424, 331079253120, 260480095349760, 278592031202284800, 388855261570122547200, 686533182382689959116800, 1495779844806108697677004800, 3942052104672989614027181260800, 12360865524060039746012601384960000

Offset: 0

Author

Alois P. Heinz, Aug 10 2014

Keywords

• nonn

Examples

			a(1) = 3: (1,1), (1,2), (2,2).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..198

Crossrefs

Cf. A241981, A246050.

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-> `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] = Which[n==0, 1, i<1, 0, True, 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_] := If[n == 0, 1, A[2n, n] - A[2n, n-1]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 01 2017, translated from Maple *)

Formula

a(n) = A241981(2n,n).

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