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A246612 Number of endofunctions on [n] whose cycle lengths are multiples of 5.

%I A246612 #11 Jun 11 2024 01:39:05
%S A246612 1,0,0,0,0,24,720,17640,430080,11022480,302835456,8976567600,
%T A246612 287236730880,9901354268520,366504424366080,14515574208519384,
%U A246612 612912858447052800,27497080096591516320,1306520704652165775360,65556843230110158804960,3464412427539012262035456
%N A246612 Number of endofunctions on [n] whose cycle lengths are multiples of 5.
%H A246612 Alois P. Heinz, <a href="/A246612/b246612.txt">Table of n, a(n) for n = 0..300</a>
%F A246612 E.g.f.: 1/(1+LambertW(-x)^5)^(1/5). - _Vaclav Kotesovec_, Sep 01 2014
%F A246612 a(n) ~ n^(n-2/5) * 2^(2/5) * (sqrt(Pi) / (5^(1/5) * Gamma(1/10))) * (1 - 14 * sqrt(2/n) * Gamma(1/10) / (75 * Gamma(3/5))). - _Vaclav Kotesovec_, Sep 01 2014
%p A246612 with(combinat):
%p A246612 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A246612       `if`(i>n, 0, add(b(n-i*j, i+5)*(i-1)!^j*
%p A246612       multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
%p A246612     end:
%p A246612 a:= a->add(b(j, 5)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246612 seq(a(n), n=0..25);
%t A246612 CoefficientList[Series[1/(1+LambertW[-x]^5)^(1/5),{x,0,20}],x] * Range[0,20]!  (* _Vaclav Kotesovec_, Sep 01 2014 *)
%Y A246612 Column k=5 of A246609.
%K A246612 nonn
%O A246612 0,6
%A A246612 _Alois P. Heinz_, Aug 31 2014