A076025 Expansion of g.f.: (1-3*x*C)/(1-4*x*C) where C = (1 - sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers A000108.

1, 1, 5, 26, 137, 726, 3858, 20532, 109361, 582782, 3106550, 16562668, 88314634, 470942044, 2511443268, 13393472616, 71428622337, 380940866574, 2031641406798, 10835261623356, 57787472903502, 308197667445204, 1643712737618748, 8766437439778776, 46754218658948922

Offset

0,3

Comments

From Paul Barry, Sep 23 2009: (Start)

The Hankel transform of this sequence is 3n+1 or 1,4,7,10,... (A016777).

The Hankel transform of the aeration of this sequence is A016777 doubled, that is, 1,1,4,4,7,7,...

In general, the Hankel transform of [x^n](1-r*xc(x))/(1-(r+1)*xc(x)) is rn+1, and that of the corresponding aerated sequence is the doubled sequence of rn+1. (End)

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

José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.

Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Catalan-Spitzer permutations, arXiv:2310.06288 [math.CO], 2023. See p. 20.

Formula

a(n+1) = Sum_{k=0..n} 3^k*binomial(2n+1, n-k)*2*(k+1)/(n+k+2). - Paul Barry, Jun 22 2004

a(n+1) = Sum_{k=0..n} A039598(n,k)*3^k. - Philippe Deléham, Mar 21 2007

a(n) = Sum_{k=0..n} A039599(n,k)*A015518(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,-4). - 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:

5, 1, 0, 0, 0, ...

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

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

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

... (End)

D-finite with recurrence: 3*n*a(n) +2*(9-14*n)*a(n-1) +32*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011

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

The sequence is the INVERT transform of A049027: (1, 4, 17, 74, 326, ...) and the third INVERT transform of the Catalan sequence (1, 2, 5, ...). - Gary W. Adamson, Jun 23 2015

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

Mathematica

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

Prog

(PARI) my(x='x+O('x^30)); Vec((1-3*sqrt(1-4*x))/(2-4*sqrt(1-4*x))) \\ G. C. Greubel, May 04 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1- 3*Sqrt(1-4*x))/(2-4*Sqrt(1-4*x)) )); // G. C. Greubel, May 04 2019
(Sage) ((1-3*sqrt(1-4*x))/(2-4*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 04 2019

Crossrefs

Cf. A000108, A001700, A049027, A076026.

Sequence in context: A018903 A355361 A083331 * A288785 A161731 A049607

Adjacent sequences: A076022 A076023 A076024 * A076026 A076027 A076028

Keyword

nonn,easy

Author

N. J. A. Sloane, Oct 29 2002

Status

approved