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

0,2

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).

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

Conjecture: 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

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 *)

Crossrefs

Cf. A000108, A000984, A007854, A076036.

Sequence in context: A155485 A155181 A082761 * A120978 A035028 A104550

Adjacent sequences: A076032 A076033 A076034 * A076036 A076037 A076038

Keyword

nonn

Author

N. J. A. Sloane, Oct 29 2002

Status

approved