A246214 Number of endofunctions on [n] where the largest cycle length equals 4.

6, 150, 3240, 72030, 1719060, 44520840, 1252364400, 38167414560, 1255558958280, 44404434904830, 1681726757430720, 67953913291104750, 2919509551303952880, 132943540577100047760, 6397727538671302783680, 324511272091351156939200, 17306903935107005765263200

Offset

4,1

Links

Alois P. Heinz, Table of n, a(n) for n = 4..200

Formula

a(n) ~ (4*exp(25/12) - 3*exp(11/6)) * 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, 4) -A(n, 3):
seq(a(n), n=4..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, 4] - A[n, 3];
a /@ Range[4, 25] (* Jean-François Alcover, Dec 28 2020, after Alois P. Heinz *)

Crossrefs

Column k=4 of A241981.

Sequence in context: A346694 A070025 A291110 * A065946 A222051 A285747

Adjacent sequences: A246211 A246212 A246213 * A246215 A246216 A246217

Keyword

nonn

Author

Alois P. Heinz, Aug 19 2014

Status

approved