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

0,3

Links

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

Formula

E.g.f.: exp(Sum_{d|8} (-LambertW(-x))^d/d).

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);

Crossrefs

Column k=8 of A246522.

Sequence in context: A268163 A050386 A246524 * A246951 A302587 A302608

Adjacent sequences: A246525 A246526 A246527 * A246529 A246530 A246531

Keyword

nonn

Author

Alois P. Heinz, Aug 28 2014

Status

approved