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A246525 Number of endofunctions on [n] whose cycle lengths are divisors of 5.

%I A246525 #5 Aug 28 2014 17:13:11
%S A246525 1,1,3,16,125,1320,17671,286336,5436153,118144000,2889312875,
%T A246525 78480441216,2343333279157,76274737767424,2687742759243375,
%U A246525 101931212748928000,4139544785141163761,179235455194948829184,8242391462093927638867,401202300756829929472000
%N A246525 Number of endofunctions on [n] whose cycle lengths are divisors of 5.
%H A246525 Alois P. Heinz, <a href="/A246525/b246525.txt">Table of n, a(n) for n = 0..350</a>
%F A246525 E.g.f.: exp(Sum_{d|5} (-LambertW(-x))^d/d).
%p A246525 with(numtheory):
%p A246525 egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
%p A246525 a:= n-> n!*coeff(series(egf(5), x, n+1), x, n):
%p A246525 seq(a(n), n=0..25);
%p A246525 # second Maple program:
%p A246525 with(combinat):
%p A246525 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A246525       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
%p A246525       (i-1)!^j, j=0..`if`(irem(5, i)=0, n/i, 0))))
%p A246525     end:
%p A246525 a:= n-> add(b(j, min(5, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246525 seq(a(n), n=0..25);
%Y A246525 Column k=5 of A246522.
%K A246525 nonn
%O A246525 0,3
%A A246525 _Alois P. Heinz_, Aug 28 2014