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A246613 Number of endofunctions on [n] whose cycle lengths are multiples of 6.

%I A246613 #11 Jun 11 2024 01:39:01
%S A246613 1,0,0,0,0,0,120,5040,161280,4898880,151200000,4870182240,
%T A246613 165589522560,5970883913280,228632817853440,9292843409850000,
%U A246613 400385970127503360,18252778818295082880,878619947953407528960,44562094805330883023040,2376296054962478899200000
%N A246613 Number of endofunctions on [n] whose cycle lengths are multiples of 6.
%H A246613 Alois P. Heinz, <a href="/A246613/b246613.txt">Table of n, a(n) for n = 0..300</a>
%F A246613 E.g.f.: 1/(1-LambertW(-x)^6)^(1/6). - _Vaclav Kotesovec_, Sep 01 2014
%F A246613 a(n) ~ n^(n-5/12) * 2^(1/4) * (sqrt(Pi) / (3^(1/6) * Gamma(1/12))) * (1 - 85 * sqrt(2/n) * Gamma(1/12) / (432 * Gamma(7/12))). - _Vaclav Kotesovec_, Sep 01 2014
%p A246613 with(combinat):
%p A246613 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A246613       `if`(i>n, 0, add(b(n-i*j, i+6)*(i-1)!^j*
%p A246613       multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
%p A246613     end:
%p A246613 a:= a->add(b(j, 6)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246613 seq(a(n), n=0..25);
%t A246613 CoefficientList[Series[1/(1-LambertW[-x]^6)^(1/6),{x,0,20}],x] * Range[0,20]!  (* _Vaclav Kotesovec_, Sep 01 2014 *)
%Y A246613 Column k=6 of A246609.
%K A246613 nonn
%O A246613 0,7
%A A246613 _Alois P. Heinz_, Aug 31 2014