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A246615 Number of endofunctions on [n] whose cycle lengths are multiples of 8.

%I A246615 #11 Jun 11 2024 01:38:31
%S A246615 1,0,0,0,0,0,0,0,5040,362880,18144000,804988800,34488115200,
%T A246615 1482082842240,65120246231040,2955402450000000,139297568464454400,
%U A246615 6837892496270035200,350034595015623598080,18692451288585606954240,1041114148173066240000000
%N A246615 Number of endofunctions on [n] whose cycle lengths are multiples of 8.
%H A246615 Alois P. Heinz, <a href="/A246615/b246615.txt">Table of n, a(n) for n = 0..300</a>
%F A246615 E.g.f.: 1/(1-LambertW(-x)^8)^(1/8). - _Vaclav Kotesovec_, Sep 01 2014
%F A246615 a(n) ~ n^(n-7/16) * 2^(1/16) * (sqrt(Pi) / Gamma(1/16)) * (1 - 161 * sqrt(2/n) * Gamma(1/16) / (768 * Gamma(9/16))). - _Vaclav Kotesovec_, Sep 01 2014
%p A246615 with(combinat):
%p A246615 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A246615       `if`(i>n, 0, add(b(n-i*j, i+8)*(i-1)!^j*
%p A246615       multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
%p A246615     end:
%p A246615 a:= a->add(b(j, 8)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246615 seq(a(n), n=0..25);
%t A246615 CoefficientList[Series[1/(1-LambertW[-x]^8)^(1/8),{x,0,20}],x] * Range[0,20]!  (* _Vaclav Kotesovec_, Sep 01 2014 *)
%Y A246615 Column k=8 of A246609.
%K A246615 nonn
%O A246615 0,9
%A A246615 _Alois P. Heinz_, Aug 31 2014