A126087 Expansion of c(2*x^2)/(1-x*c(2*x^2)), where c(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108).

1, 1, 3, 5, 15, 29, 87, 181, 543, 1181, 3543, 7941, 23823, 54573, 163719, 381333, 1143999, 2699837, 8099511, 19319845, 57959535, 139480397, 418441191, 1014536117, 3043608351, 7426790749, 22280372247, 54669443141, 164008329423

Offset

0,3

Comments

Series reversion of x*(1+x)/(1+2*x+3*x^2) [offset 0]. - Paul Barry, Mar 13 2007

Hankel transform is 2^C(n+1,2). - Philippe Deléham, Mar 16 2007

Links

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

Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.

Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.

Formula

G.f.: (1-sqrt(1-8*x^2))/(x*(4*x-1+sqrt(1-8*x^2))). - Emeric Deutsch, Mar 04 2007

a(n) = Sum_{k=0..n} 2^(n-k)*A120730(n,k). - Philippe Deléham, Oct 16 2008

a(n) = Sum_(k=1..n} (1+(-1)^(n-k))*k*2^((n-k)/2-1)*C(n,(n+k)/2)/n, n>0. - Vladimir Kruchinin, Feb 18 2011

a(2*n) = A089022(n). - Philippe Deléham, Nov 02 2011

D-finite with recurrence: (n+1)*a(n) = 3*(n+1)*a(n-1) - 8*(2-n)*a(n-2) - 24*(n-2)*a(n-3). - R. J. Mathar, Nov 14 2011

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

Maple

c:=x->(1-sqrt(1-4*x))/2/x: G:=c(2*x^2)/(1-x*c(2*x^2)): Gser:=series(G, x=0, 35): seq(coeff(Gser, x, n), n=0..32); # Emeric Deutsch, Mar 04 2007

Mathematica

CoefficientList[Series[(1-Sqrt[1-8*x^2])/(x*(4*x-1+Sqrt[1-8*x^2])), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-8*x^2))/(x*(4*x-1+Sqrt(1-8*x^2))) )); // G. C. Greubel, Nov 07 2022
(SageMath)
def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A126087(n): return sum(2^(n-k)*A120730(n, k) for k in range(n+1))
[A126087(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

Crossrefs

Cf. A000108, A089022, A120730.

Sequence in context: A166956 A048738 A018454 * A148498 A259921 A292689

Adjacent sequences: A126084 A126085 A126086 * A126088 A126089 A126090

Keyword

nonn

Author

Philippe Deléham, Mar 03 2007

Extensions

More terms from Emeric Deutsch, Mar 04 2007

Status

approved