A246191 Number of endofunctions on [n] where the smallest cycle length equals 4.

6, 120, 2160, 41160, 861420, 19949328, 510320160, 14348862000, 440879024520, 14716697990280, 530761366078944, 20577610843203960, 853717568817968400, 37746072677473752480, 1771994498414094109440, 88032162789004128733152, 4614300279345812506938720

Offset

4,1

Links

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

Formula

a(n) ~ (exp(-11/6) - exp(-25/12)) * 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, 4) -A(n, 5):
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>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, 4] - A[n, 5];
a /@ Range[4, 25] (* Jean-François Alcover, Dec 28 2020, after Alois P. Heinz *)

Crossrefs

Column k=4 of A246049.

Sequence in context: A026337 A223629 A065888 * A246611 A185757 A075844

Adjacent sequences: A246188 A246189 A246190 * A246192 A246193 A246194

Keyword

nonn

Author

Alois P. Heinz, Aug 18 2014

Status

approved