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A246610 Number of endofunctions on [n] whose cycle lengths are multiples of 3.

%I A246610 #12 Jun 11 2024 01:39:13
%S A246610 1,0,0,2,24,300,4480,78750,1591296,36355256,927244800,26127386010,
%T A246610 806251494400,27046291980708,980094896062464,38158333538165750,
%U A246610 1588601646620835840,70427042234715548400,3312574102411273437184,164767312911755127462066,8641342923227371929600000
%N A246610 Number of endofunctions on [n] whose cycle lengths are multiples of 3.
%H A246610 Alois P. Heinz, <a href="/A246610/b246610.txt">Table of n, a(n) for n = 0..300</a>
%F A246610 E.g.f.: 1/(1+LambertW(-x)^3)^(1/3). - _Vaclav Kotesovec_, Sep 01 2014
%F A246610 a(n) ~ Gamma(5/6) * (n^(n-1/3) / (12^(1/3) * sqrt(Pi))) * (1 - 2^(7/6) * Gamma(1/3)^3 / (9 * Pi^(3/2) * sqrt(n))). - _Vaclav Kotesovec_, Sep 01 2014
%p A246610 with(combinat):
%p A246610 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A246610       `if`(i>n, 0, add(b(n-i*j, i+3)*(i-1)!^j*
%p A246610       multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
%p A246610     end:
%p A246610 a:= a->add(b(j, 3)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246610 seq(a(n), n=0..20);
%t A246610 CoefficientList[Series[1/(1+LambertW[-x]^3)^(1/3),{x,0,20}],x] * Range[0,20]!  (* _Vaclav Kotesovec_, Sep 01 2014 *)
%Y A246610 Column k=3 of A246609.
%K A246610 nonn
%O A246610 0,4
%A A246610 _Alois P. Heinz_, Aug 31 2014