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A246616 Number of endofunctions on [n] whose cycle lengths are multiples of 9.

%I A246616 #11 Jun 11 2024 01:38:28
%S A246616 1,0,0,0,0,0,0,0,0,40320,3628800,219542400,11496038400,570031862400,
%T A246616 27908676956160,1379187810000000,69648048665395200,
%U A246616 3619848890071814400,194428045753727385600,10817313676715756770560,624148118196535296000000,37365232440634029661766400
%N A246616 Number of endofunctions on [n] whose cycle lengths are multiples of 9.
%H A246616 Alois P. Heinz, <a href="/A246616/b246616.txt">Table of n, a(n) for n = 0..300</a>
%F A246616 E.g.f.: 1/(1+LambertW(-x)^9)^(1/9). - _Vaclav Kotesovec_, Sep 01 2014
%F A246616 a(n) ~ n^(n-4/9) * 2^(4/9) * sqrt(Pi) / (3^(2/9) * Gamma(1/18)) * (1 - 52 * sqrt(2/n) * Gamma(1/18) / (243 * Gamma(5/9))). - _Vaclav Kotesovec_, Sep 01 2014
%p A246616 with(combinat):
%p A246616 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A246616       `if`(i>n, 0, add(b(n-i*j, i+9)*(i-1)!^j*
%p A246616       multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
%p A246616     end:
%p A246616 a:= a->add(b(j, 9)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246616 seq(a(n), n=0..25);
%t A246616 CoefficientList[Series[1/(1+LambertW[-x]^9)^(1/9),{x,0,20}],x] * Range[0,20]!  (* _Vaclav Kotesovec_, Sep 01 2014 *)
%Y A246616 Column k=9 of A246609.
%K A246616 nonn
%O A246616 0,10
%A A246616 _Alois P. Heinz_, Aug 31 2014