A246218 Number of endofunctions on [n] where the largest cycle length equals 8.

5040, 408240, 22680000, 1106859600, 51732172800, 2408384618640, 113960430904320, 5541379593750000, 278592031202284800, 14529619059476320800, 787422201081850414080, 44373594768472437642720, 2600326096882824360960000, 158404803877320370312773600

Offset: 8

Author

Alois P. Heinz, Aug 19 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 8..200

Crossrefs

Column k=8 of A241981.

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-> A(n, 8) -A(n, 7):
seq(a(n), n=8..25);

Formula

a(n) ~ (8*exp(761/280) - 7*exp(363/140)) * n^(n-1). - Vaclav Kotesovec, Aug 21 2014