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

Author

Philippe Deléham, Mar 03 2007

Keywords

• nonn

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.

Crossrefs

Cf. A000108, A089022, A120730.

Programs

Magma

R:=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

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 *)

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

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-1) = Sum_{k=1..n} (1+(-1)^(n-k))*k*2^((n-k)/2-1)*C(n,floor((n+k)/2))/n. - 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

Extensions

More terms from Emeric Deutsch, Mar 04 2007