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A246528 Number of endofunctions on [n] whose cycle lengths are divisors of 8.

%I A246528 #5 Aug 28 2014 20:35:16
%S A246528 1,1,4,25,224,2601,37072,626137,12232320,271494865,6750538496,
%T A246528 185923318329,5619645500416,184961854976185,6585429015521280,
%U A246528 252203521861645561,10338251689510381568,451650823526438037153,20949317446607098716160,1028215744082428119960025
%N A246528 Number of endofunctions on [n] whose cycle lengths are divisors of 8.
%H A246528 Alois P. Heinz, <a href="/A246528/b246528.txt">Table of n, a(n) for n = 0..350</a>
%F A246528 E.g.f.: exp(Sum_{d|8} (-LambertW(-x))^d/d).
%p A246528 with(numtheory):
%p A246528 egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
%p A246528 a:= n-> n!*coeff(series(egf(8), x, n+1), x, n):
%p A246528 seq(a(n), n=0..25);
%p A246528 # second Maple program:
%p A246528 with(combinat):
%p A246528 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A246528       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
%p A246528       (i-1)!^j, j=0..`if`(irem(8, i)=0, n/i, 0))))
%p A246528     end:
%p A246528 a:= n-> add(b(j, min(8, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246528 seq(a(n), n=0..25);
%Y A246528 Column k=8 of A246522.
%K A246528 nonn
%O A246528 0,3
%A A246528 _Alois P. Heinz_, Aug 28 2014