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A246524 Number of endofunctions on [n] whose cycle lengths are divisors of 4.

%I A246524 #5 Aug 28 2014 17:00:45
%S A246524 1,1,4,25,224,2601,37072,626137,12227280,271086625,6727858496,
%T A246524 184818121929,5568152828416,182575550335465,6473161538599680,
%U A246524 246781048203043561,10067677495565927168,437653901985319521153,20197310874805488471040,986221173076368356013625
%N A246524 Number of endofunctions on [n] whose cycle lengths are divisors of 4.
%H A246524 Alois P. Heinz, <a href="/A246524/b246524.txt">Table of n, a(n) for n = 0..350</a>
%F A246524 E.g.f.: exp(Sum_{d|4} (-LambertW(-x))^d/d).
%p A246524 with(numtheory):
%p A246524 egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
%p A246524 a:= n-> n!*coeff(series(egf(4), x, n+1), x, n):
%p A246524 seq(a(n), n=0..25);
%p A246524 # second Maple program:
%p A246524 with(combinat):
%p A246524 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A246524       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
%p A246524       (i-1)!^j, j=0..`if`(irem(4, i)=0, n/i, 0))))
%p A246524     end:
%p A246524 a:= n-> add(b(j, min(4, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246524 seq(a(n), n=0..25);
%Y A246524 Column k=4 of A246522.
%K A246524 nonn
%O A246524 0,3
%A A246524 _Alois P. Heinz_, Aug 28 2014