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A246611 Number of endofunctions on [n] whose cycle lengths are multiples of 4.

%I A246611 #11 Jun 11 2024 01:39:08
%S A246611 1,0,0,0,6,120,2160,41160,866460,20294064,526680000,15036999120,
%T A246611 468848156040,15859299473160,578619457031616,22654279249875000,
%U A246611 947570269816868880,42174922731482980320,1990416896317283627520,99290011292792071612704,5220362654145754082460000
%N A246611 Number of endofunctions on [n] whose cycle lengths are multiples of 4.
%H A246611 Alois P. Heinz, <a href="/A246611/b246611.txt">Table of n, a(n) for n = 0..300</a>
%F A246611 E.g.f.: 1/(1-LambertW(-x)^4)^(1/4). - _Vaclav Kotesovec_, Sep 01 2014
%F A246611 a(n) ~ n^(n-3/8) * (sqrt(Pi) / (2^(1/8) * Gamma(1/8))) * (1 - 11 * sqrt(2/n) * Gamma(1/8) / (64 * Gamma(5/8))). - _Vaclav Kotesovec_, Sep 01 2014
%p A246611 with(combinat):
%p A246611 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A246611       `if`(i>n, 0, add(b(n-i*j, i+4)*(i-1)!^j*
%p A246611       multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
%p A246611     end:
%p A246611 a:= a->add(b(j, 4)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246611 seq(a(n), n=0..20);
%t A246611 CoefficientList[Series[1/(1-LambertW[-x]^4)^(1/4),{x,0,20}],x] * Range[0,20]!  (* _Vaclav Kotesovec_, Sep 01 2014 *)
%Y A246611 Column k=4 of A246609.
%K A246611 nonn
%O A246611 0,5
%A A246611 _Alois P. Heinz_, Aug 31 2014