A092807 Expansion of (1-6*x+4*x^2)/((1-2*x)*(1-6*x)).

1, 2, 8, 40, 224, 1312, 7808, 46720, 280064, 1679872, 10078208, 60467200, 362799104, 2176786432, 13060702208, 78364180480, 470185017344, 2821109972992, 16926659575808, 101559956930560, 609359740534784

Offset: 0

Author

Paul Barry, Mar 06 2004

Keywords

• easy
• nonn

Comments

Second binomial transform of A054881 (closed walks at a vertex of an octahedron) With interpolated zeros, counts closed walks of length n at a vertex of the edge-vertex incidence graph of K_4 associated with the edges of K_4.

This also gives the number of noncrossing, nonnesting, 2-colored permutations on {1, 2, ..., n}. - Lily Yen, Apr 22 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Lily Yen, Crossings and Nestings for Arc-Coloured Permutations, arXiv:1211.3472 [math.CO], 2012-2013 and Arc-coloured permutations, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754.
• Lily Yen, Crossings and Nestings for Arc-Coloured Permutations and Automation, Electronic Journal of Combinatorics, 22(1) (2015), #P1.14.
• Index entries for linear recurrences with constant coefficients, signature (8,-12).

Crossrefs

Cf. A054881, A074601, A092803, A124302.

Programs

Magma

[1] cat [6^(n-1) + 2^(n-1): n in [1..40]]; // G. C. Greubel, Jan 04 2023

Mathematica

CoefficientList[Series[(1-6x+4x^2)/((1-2x)(1-6x)),{x,0,40}],x] (* or *) LinearRecurrence[{8,-12},{1,2,8},41] (* Harvey P. Dale, Aug 23 2011 *)

SageMath

[(6^n + 3*2^n + 2*0^n)/6 for n in range(41)] # G. C. Greubel, Jan 04 2023

Formula

G.f.: (1-6*x+4*x^2)/((1-2*x)*(1-6*x)).

a(n) = (6^n + 3*2^n + 2*0^n)/6.

a(n) = A074601(n-1), n>0. - R. J. Mathar, Sep 08 2008

a(0)=1, a(1)=2, a(2)=8, a(n) = 8*a(n-1)-12*a(n-2). - Harvey P. Dale, Aug 23 2011

a(n) = A124302(n)*2^n. - Philippe Deléham, Nov 01 2011

E.g.f.: (1/6)*( 1 + 3*exp(2*x) + exp(6*x) ). - G. C. Greubel, Jan 04 2023