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

%I A246523 #6 Aug 28 2014 16:59:11
%S A246523 1,1,3,18,157,1776,24687,407464,7792857,169554240,4137133051,
%T A246523 111912543744,3324740466357,107628168419968,3771341043102375,
%U A246523 142230049514309376,5744687204783023153,247424591909961916416,11320453594446364577907,548348501001426735001600
%N A246523 Number of endofunctions on [n] whose cycle lengths are divisors of 3.
%H A246523 Alois P. Heinz, <a href="/A246523/b246523.txt">Table of n, a(n) for n = 0..350</a>
%F A246523 E.g.f.: exp(Sum_{d|3} (-LambertW(-x))^d/d).
%p A246523 with(numtheory):
%p A246523 egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
%p A246523 a:= n-> n!*coeff(series(egf(3), x, n+1), x, n):
%p A246523 seq(a(n), n=0..25);
%p A246523 # second Maple program:
%p A246523 with(combinat):
%p A246523 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A246523       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
%p A246523       (i-1)!^j, j=0..`if`(irem(3, i)=0, n/i, 0))))
%p A246523     end:
%p A246523 a:= n-> add(b(j, min(3, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246523 seq(a(n), n=0..25);
%Y A246523 Column k=3 of A246522.
%K A246523 nonn
%O A246523 0,3
%A A246523 _Alois P. Heinz_, Aug 28 2014