A246190 Number of endofunctions on [n] where the smallest cycle length equals 3.

2, 24, 300, 4360, 74130, 1456224, 32562152, 817596000, 22785399450, 697951656160, 23306666102148, 842567564800416, 32781106696806650, 1365579024023558400, 60639189588419033040, 2859165143013913590016, 142651621238828972159538, 7508140027468431374563200

Offset: 3

Author

Alois P. Heinz, Aug 18 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..200

Crossrefs

Column k=3 of A246049.

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

Formula

a(n) ~ (exp(-3/2) - exp(-11/6)) * n^n. - Vaclav Kotesovec, Aug 21 2014