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

3,1

Links

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

Formula

a(n) ~ (exp(-3/2) - exp(-11/6)) * 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, 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 *)

Crossrefs

Column k=3 of A246049.

Sequence in context: A135389 A336310 A065513 * A246610 A119491 A001864

Adjacent sequences: A246187 A246188 A246189 * A246191 A246192 A246193

Keyword

nonn

Author

Alois P. Heinz, Aug 18 2014

Status

approved