A076035 G.f.: 1/(1-4*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

1, 4, 20, 104, 548, 2904, 15432, 82128, 437444, 2331128, 12426200, 66250672, 353258536, 1883768176, 10045773072, 53573890464, 285714489348, 1523763466296, 8126565627192, 43341046493424, 231149891614008, 1232790669780816, 6574850950474992, 35065749759115104

Offset: 0

Author

N. J. A. Sloane, Oct 29 2002

Keywords

• nonn

Comments

The Hankel transform of this sequence and that of the aerated sequence with g.f. 1/(1-4x^2*c(x^2)) is 4^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n. - Paul Barry, Jan 20 2007

Links

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

Crossrefs

Cf. A000108, A000984, A007854, A076036.

Programs

Maple

CatalanNumber := n -> binomial(2*n,n)/(n+1):
h := (n, m) -> hypergeom([1+m, m-n], [m+n+2], -3):
a := n -> CatalanNumber(n)*(h(n,0) + 6*n/(n+2)*h(n,1)):
seq(simplify(a(n)), n=0..23); # Peter Luschny, Dec 09 2018

Mathematica

CoefficientList[Series[1/(1-4*x*(1-Sqrt[1-4*x])/(2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

Formula

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

a(n) = Sum_{k, 0<=k<=n} A106566(n, k)*4^k. - Philippe Deléham, Sep 01 2005

a(n) = if(n=0,1,sum{k=1..n, C(2n-k-1,n-k)*k*4^k/n}). - Paul Barry, Jan 20 2007

a(n) = Sum{k, 0<=k<=n}A039599(n,k)*3^k. - Philippe Deléham, Sep 08 2007

a(0)=1, a(n)=(16*a(n-1)-4*A000108(n-1))/3. - Philippe Deléham, Nov 27 2007

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 [proved by Ekhad & Yang, see link]

a(n) ~ 2^(4*n+1) / 3^(n+1). - Vaclav Kotesovec, Feb 13 2014

Conjecture: a(n) = 4*A076025(n), n>0. - R. J. Mathar, Apr 01 2022