A246610 - OEIS

%I #12 Jun 11 2024 01:39:13

%S 1,0,0,2,24,300,4480,78750,1591296,36355256,927244800,26127386010,

%T 806251494400,27046291980708,980094896062464,38158333538165750,

%U 1588601646620835840,70427042234715548400,3312574102411273437184,164767312911755127462066,8641342923227371929600000

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

%H Alois P. Heinz, <a href="/A246610/b246610.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f.: 1/(1+LambertW(-x)^3)^(1/3). - _Vaclav Kotesovec_, Sep 01 2014

%F 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 with(combinat):

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p       `if`(i>n, 0, add(b(n-i*j, i+3)*(i-1)!^j*

%p       multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))

%p     end:

%p a:= a->add(b(j, 3)*n^(n-j)*binomial(n-1, j-1), j=0..n):

%p seq(a(n), n=0..20);

%t CoefficientList[Series[1/(1+LambertW[-x]^3)^(1/3),{x,0,20}],x] * Range[0,20]!  (* _Vaclav Kotesovec_, Sep 01 2014 *)

%Y Column k=3 of A246609.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Aug 31 2014