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

%I #6 Aug 28 2014 16:59:11

%S 1,1,3,18,157,1776,24687,407464,7792857,169554240,4137133051,

%T 111912543744,3324740466357,107628168419968,3771341043102375,

%U 142230049514309376,5744687204783023153,247424591909961916416,11320453594446364577907,548348501001426735001600

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

%H Alois P. Heinz, <a href="/A246523/b246523.txt">Table of n, a(n) for n = 0..350</a>

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

%p with(numtheory):

%p egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):

%p a:= n-> n!*coeff(series(egf(3), x, n+1), x, n):

%p seq(a(n), n=0..25);

%p # second Maple program:

%p with(combinat):

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*

%p (i-1)!^j, j=0..`if`(irem(3, i)=0, n/i, 0))))

%p end:

%p a:= n-> add(b(j, min(3, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):

%p seq(a(n), n=0..25);

%Y Column k=3 of A246522.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 28 2014