A246611 - OEIS

%I #11 Jun 11 2024 01:39:08

%S 1,0,0,0,6,120,2160,41160,866460,20294064,526680000,15036999120,

%T 468848156040,15859299473160,578619457031616,22654279249875000,

%U 947570269816868880,42174922731482980320,1990416896317283627520,99290011292792071612704,5220362654145754082460000

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

%H Alois P. Heinz, <a href="/A246611/b246611.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f.: 1/(1-LambertW(-x)^4)^(1/4). - _Vaclav Kotesovec_, Sep 01 2014

%F 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 with(combinat):

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p       `if`(i>n, 0, add(b(n-i*j, i+4)*(i-1)!^j*

%p       multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))

%p     end:

%p a:= a->add(b(j, 4)*n^(n-j)*binomial(n-1, j-1), j=0..n):

%p seq(a(n), n=0..20);

%t CoefficientList[Series[1/(1-LambertW[-x]^4)^(1/4),{x,0,20}],x] * Range[0,20]!  (* _Vaclav Kotesovec_, Sep 01 2014 *)

%Y Column k=4 of A246609.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Aug 31 2014