A246526 Number of endofunctions on [n] whose cycle lengths are divisors of 6.

1, 1, 4, 27, 250, 2951, 42552, 726097, 14318908, 320511105, 8029282096, 222590246099, 6765751467576, 223748991426247, 7998566722112800, 307359039816710361, 12634664945078752528, 553260940314226017473, 25711427896197877574208, 1263904006537455579001675

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=6 of A246522.

Programs

Maple

with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(6), 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(6, i)=0, n/i, 0))))
    end:
a:= n-> add(b(j, min(6, 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|6} (-LambertW(-x))^d/d).