A097632 - OEIS

%I #27 Jan 16 2024 10:30:54

%S 2,12,64,672,8448,138240,2672640,60641280,1568931840,45705461760,

%T 1478924697600,52646746521600,2044394156851200,86005817907609600,

%U 3896481847600742400,189139342470414336000,9793081532749971456000,538748376721309827072000,31381673358053118836736000

%N a(n) = 2^n * Lucas(n) * (n-1)!.

%C Number of possible well-colored cycles on n nodes. Well-colored means, each green vertex has at least a red child, each red vertex has no red child.

%H A.H.M. Smeets, <a href="/A097632/b097632.txt">Table of n, a(n) for n = 1..100</a>

%H C. Banderier, J.-M. Le Bars, and V. Ravelomanana, <a href="https://arxiv.org/abs/math/0411138">Generating functions for kernels of digraphs</a>, arXiv:math/0411138 [math.CO], 2004.

%F E.g.f.: -log(1-2*x-4*x^2).

%F a(n) = A000204(n) * A066318(n).

%F a(n) ~ sqrt(2*Pi/n)*(2*n*phi/e)^n. - _Stefano Spezia_, Jan 16 2024

%t a[n_] := 2^n*LucasL[n,1]*(n-1)!; Array[a,19] (* or *)

%t nmax=19; CoefficientList[Series[-Log[1-2x-4x^2], {x,0,nmax}], x]Range[0,nmax]! (* _Stefano Spezia_, Jan 15 2024 *)

%o (Python)

%o def A097632(n):

%o     L0, L1, F, i = 1, 2, 2, 1

%o     while i < n:

%o         L0, L1, F, i = L0+L1, L0, 2*i*F, i+1

%o     return L0*F # _A.H.M. Smeets_, Jan 15 2024

%Y Cf. A000079, A000142, A000204, A066318, A087131.

%K nonn

%O 1,1

%A _Ralf Stephan_, Aug 17 2004

%E Definition corrected by and a(18)-a(19) from _Stefano Spezia_, Jan 15 2024