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A246529 Number of endofunctions on [n] whose cycle lengths are divisors of 9.

%I A246529 #8 Oct 26 2014 17:30:13
%S A246529 1,1,3,18,157,1776,24687,407464,7792857,169594560,4141165051,
%T A246529 112178655744,3339749183157,108422228887168,3812520677598375,
%U A246529 144372964560581376,5858088633723823153,253575577033176047616,11664031615012086920307,568166632439929892761600
%N A246529 Number of endofunctions on [n] whose cycle lengths are divisors of 9.
%H A246529 Alois P. Heinz, <a href="/A246529/b246529.txt">Table of n, a(n) for n = 0..350</a>
%F A246529 E.g.f.: exp(Sum_{d|9} (-LambertW(-x))^d/d).
%F A246529 a(n) - A246523(n) is a multiple of 40320. - _M. F. Hasler_, Oct 26 2014
%p A246529 with(numtheory):
%p A246529 egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
%p A246529 a:= n-> n!*coeff(series(egf(9), x, n+1), x, n):
%p A246529 seq(a(n), n=0..25);
%p A246529 # second Maple program:
%p A246529 with(combinat):
%p A246529 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A246529       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
%p A246529       (i-1)!^j, j=0..`if`(irem(9, i)=0, n/i, 0))))
%p A246529     end:
%p A246529 a:= n-> add(b(j, min(9, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246529 seq(a(n), n=0..25);
%Y A246529 Column k=9 of A246522.
%K A246529 nonn
%O A246529 0,3
%A A246529 _Alois P. Heinz_, Aug 28 2014