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

0,3

Links

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

Formula

E.g.f.: exp(Sum_{d|3} (-LambertW(-x))^d/d).

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);

Crossrefs

Column k=3 of A246522.

Sequence in context: A138274 A181374 A060913 * A246529 A305824 A116956

Adjacent sequences: A246520 A246521 A246522 * A246524 A246525 A246526

Keyword

nonn

Author

Alois P. Heinz, Aug 28 2014

Status

approved