A076026 Expansion of g.f.: (1-4*x*C)/(1-5*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

1, 1, 6, 37, 230, 1434, 8952, 55917, 349374, 2183230, 13643972, 85270626, 532926716, 3330739972, 20816939100, 130105200765, 813155081070, 5082210417270, 31763782696740, 198523522444950, 1240771573465140, 7754820693127020, 48467623215477120, 302922622226091090

Offset: 0

Author

N. J. A. Sloane, Oct 29 2002

Keywords

• nonn
• easy

Comments

a(n) is the number of Motzkin paths of length n-1 in which the (1,0)-steps at level 0 come in 6 colors and those at a higher level come in 2 colors. Example: a(4)=230 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 6^3 = 216 paths of shape HHH, 6 paths of shape HUD, 6 paths of shape UDH, and 2 paths of shape UHD. - Emeric Deutsch, May 02 2011

References

• L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Catalan-Spitzer permutations, arXiv:2310.06288 [math.CO], 2023. See p. 20.

Crossrefs

Cf. A000108, A001700, A049027, A076025.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (2- 4*Sqrt(1-4*x))/(3-5*Sqrt(1-4*x)) )); // G. C. Greubel, May 04 2019

Mathematica

CoefficientList[Series[(2-4*Sqrt[1-4*x])/(3-5*Sqrt[1-4*x]), {x, 0, 30}], x] (* Vaclav Kotesovec, Dec 09 2013 *)
Flatten[{1,Table[FullSimplify[(2*n)!*Hypergeometric2F1Regularized[1, n+1/2, n+2, 16/25] / (25*n!) + 3*5^(2*n-1)/4^(n+1)], {n,1,30}]}] (* Vaclav Kotesovec, Dec 09 2013 *)

PARI

my(x='x+O('x^30)); Vec((2-4*sqrt(1-4*x))/(3-5*sqrt(1-4*x))) \\ G. C. Greubel, May 04 2019

Sage

((2-4*sqrt(1-4*x))/(3-5*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 04 2019

Formula

a(n+1) = Sum_{k=0..n} A039598(n,k)*4^k. - Philippe Deléham, Mar 21 2007

a(n) = Sum_{k=0..n} A039599(n,k)*A015521(k), for n >= 1. - Philippe Deléham, Nov 22 2007

Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n >= 1, a(n+1)=(-1)^n*charpoly(A,-5). - Milan Janjic, Jul 08 2010

From Gary W. Adamson, Jul 25 2011: (Start)

a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows:

6, 1, 0, 0, 0, ...

1, 1, 1, 0, 0, ...

1, 1, 1, 1, 0, ...

1, 1, 1, 1, 1, ...

... (End)

D-finite with recurrence: 4*n*a(n) = (41*n-24)*a(n-1) - 50*(2*n-3)*a(n-2). - Vaclav Kotesovec, Dec 09 2013

a(n) ~ 3*5^(2*n-1)/4^(n+1). - Vaclav Kotesovec, Dec 09 2013

O.g.f. A(x) = (1 - *Sum_{n >= 1} binomial(2*n,n)*x^n)/(1 - (3/2)*Sum_{n >= 1} binomial(2*n,n)*x^n). - Peter Bala, Sep 01 2016