A246523 Number of endofunctions on [n] whose cycle lengths are divisors of 3.

1, 1, 3, 18, 157, 1776, 24687, 407464, 7792857, 169554240, 4137133051, 111912543744, 3324740466357, 107628168419968, 3771341043102375, 142230049514309376, 5744687204783023153, 247424591909961916416, 11320453594446364577907, 548348501001426735001600

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=3 of A246522.

Programs

Maple

with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(3), 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(3, i)=0, n/i, 0))))
    end:
a:= n-> add(b(j, min(3, 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|3} (-LambertW(-x))^d/d).