A097632 a(n) = 2^n * Lucas(n) * (n-1)!.

2, 12, 64, 672, 8448, 138240, 2672640, 60641280, 1568931840, 45705461760, 1478924697600, 52646746521600, 2044394156851200, 86005817907609600, 3896481847600742400, 189139342470414336000, 9793081532749971456000, 538748376721309827072000, 31381673358053118836736000

Offset: 1

Author

Ralf Stephan, Aug 17 2004

Keywords

• nonn

Comments

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.

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..100
• C. Banderier, J.-M. Le Bars, and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.

Crossrefs

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

Programs

Mathematica

a[n_] := 2^n*LucasL[n,1]*(n-1)!; Array[a,19] (* or *)
nmax=19; CoefficientList[Series[-Log[1-2x-4x^2], {x,0,nmax}], x]Range[0,nmax]! (* Stefano Spezia, Jan 15 2024 *)

Python

def A097632(n):
    L0, L1, F, i = 1, 2, 2, 1
    while i < n:
        L0, L1, F, i = L0+L1, L0, 2*i*F, i+1
    return L0*F # A.H.M. Smeets, Jan 15 2024

Formula

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

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

a(n) ~ sqrt(2*Pi/n)*(2*n*phi/e)^n. - Stefano Spezia, Jan 16 2024

Extensions

Definition corrected by and a(18)-a(19) from Stefano Spezia, Jan 15 2024