A246528 Number of endofunctions on [n] whose cycle lengths are divisors of 8.

1, 1, 4, 25, 224, 2601, 37072, 626137, 12232320, 271494865, 6750538496, 185923318329, 5619645500416, 184961854976185, 6585429015521280, 252203521861645561, 10338251689510381568, 451650823526438037153, 20949317446607098716160, 1028215744082428119960025

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=8 of A246522.

Programs

Maple

with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(8), 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(8, i)=0, n/i, 0))))
    end:
a:= n-> add(b(j, min(8, 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|8} (-LambertW(-x))^d/d).