A246529 Number of endofunctions on [n] whose cycle lengths are divisors of 9.

1, 1, 3, 18, 157, 1776, 24687, 407464, 7792857, 169594560, 4141165051, 112178655744, 3339749183157, 108422228887168, 3812520677598375, 144372964560581376, 5858088633723823153, 253575577033176047616, 11664031615012086920307, 568166632439929892761600

Offset: 0

Author

Alois P. Heinz, Aug 28 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350

Crossrefs

Column k=9 of A246522.

Programs

Maple

with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(9), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
      (i-1)!^j, j=0..`if`(irem(9, i)=0, n/i, 0))))
    end:
a:= n-> add(b(j, min(9, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);

Formula

E.g.f.: exp(Sum_{d|9} (-LambertW(-x))^d/d).

a(n) - A246523(n) is a multiple of 40320. - M. F. Hasler, Oct 26 2014