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A049027 G.f.: (1-2*x*C)/(1-3*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x is the g.f. for Catalan numbers A000108. 18
1, 1, 4, 17, 74, 326, 1446, 6441, 28770, 128750, 576944, 2587850, 11615932, 52167688, 234383146, 1053386937, 4735393794, 21291593238, 95747347176, 430624242942, 1936925461644, 8712882517188, 39195738193836, 176335080590442 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row sums of triangle A035324.

[a(n+1)] = [1,4,17,74,326,...] is the binomial transform of A059738. [From Philippe Deléham, Nov 26 2009]

(1, 4, 17, 74, 326, ...) is the invert transform of the odd-indexed central binomial coefficients, A001700. [David Callan, Oct 14 2012]

The sequence starting with index 1 is the INVERT transform of A001700: (1, 3, 10, 35, 126,...) and the second INVERT transform of the Catalan numbers starting with index 1: (1, 2, 5, 14, 42,...). - Gary W. Adamson, Jun 23 2015

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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 and arXiv:1505.05568

Paul Barry, Arnauld Mesinga Mwafise, Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.

S. B. Ekhad, M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

FORMULA

G.f.: x*c(x)/(1-3*x*c(x)), c(x)= g.f. of Catalan numbers A000108.

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

a(n) = (9*a(n-1)-Catalan(n-1))/2, n>1. - Vladeta Jovovic, Aug 08 2004

a(n+1) = Sum_{k, 0<=k<=n}A039598(n,k)*2^k . - Philippe Deléham, Mar 21 2007

G.f.: 2 / (3 - 1 / sqrt(1 - 4*x)). - Michael Somos, Apr 08 2007

a(n) = Sum_{k, 0<=k<=n}A039599(n,k)*A001045(k), for n>=1 . - Philippe Deléham, Jun 10 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,-3). [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:

4, 1, 0, 0, 0,...

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

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

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

... (End)

Conjecture: 2*n*a(n) +(12-17*n)*a(n-1) +18*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011

a(n) ~ 3^(2*n-1)/2^(n+1). - Vaclav Kotesovec, Oct 08 2012

0 = a(n)*(1296*a(n+1) - 1098*a(n+2) + 180*a(n+3)) + a(n+1)*(-126*a(n+1) + 253*a(n+2) - 58*a(n+3)) + a(n+2)*(-10*a(n+2) + 4*a(n+3)) if n>0. - Michael Somos, Jan 23 2014

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

a(n) = 3^(2*n-1)/2^(n+1) + 2^n * (2*n-1)!! * hypergeom([1,n+1], [n+2], 8/9)/(9*(n+1)!) + 0^n * 2/3. - Vladimir Reshetnikov, Oct 08 2016

EXAMPLE

G.f. = 1 + x + 4*x^2 + 17*x^3 + 74*x^4 + 326*x^5 + 1446*x^6 + 6441*x^7 + ...

MATHEMATICA

Table[SeriesCoefficient[2/(3-1/Sqrt[1-4*x]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2012 *)

FunctionExpand@Table[3^(2 n - 1)/2^(n + 1) + 2^n (2 n - 1)!! Hypergeometric2F1[1, n + 1/2, n + 2, 8/9]/(9 (n + 1)!) + 2 KroneckerDelta[n]/3, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 08 2016 *)

PROG

(PARI) {a(n) = if( n<1, n==0, polcoeff( serreverse( x * (1 + 2*x) / (1 + 3*x)^2 + x * O(x^n) ), n))}; /* Michael Somos, Apr 08 2007 */

(PARI) {a(n) = if( n<0, 0, polcoeff( 2 / (3 - 1 / sqrt(1 - 4*x + x * O(x^n))), n))}; /* Michael Somos, Apr 08 2007 */

(MAGMA) [1] cat [n eq 1 select 1 else (9*Self(n-1)-Catalan(n-1))/2: n in [1..30]]; // Vincenzo Librandi, Jun 25 2015

CROSSREFS

Cf. A000108, A001700.

Sequence in context: A184700 A125586 A086351 * A026751 A227504 A218984

Adjacent sequences:  A049024 A049025 A049026 * A049028 A049029 A049030

KEYWORD

easy,nonn

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified August 17 09:55 EDT 2018. Contains 313814 sequences. (Running on oeis4.)