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
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
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Oct 29 2002
STATUS
approved