A128386 Expansion of c(3*x^2)/(1-x*c(3*x^2)), c(x) the g.f. of A000108.

1, 1, 4, 7, 28, 58, 232, 523, 2092, 4966, 19864, 48838, 195352, 492724, 1970896, 5068915, 20275660, 52955950, 211823800, 560198962, 2240795848, 5987822380, 23951289520, 64563867454, 258255469816, 701383563388, 2805534253552

Offset: 0

Author

Paul Barry, Feb 28 2007

Keywords

• easy
• nonn

Comments

Hankel transform is 3^C(n+1,2) = A047656(n+1).

Series reversion of x*(1+x)/(1+2*x+4*x^2).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Slides, 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, A009766, A047656, A120730.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (Sqrt(1-12*x^2)+2*x-1)/(2*x*(1-4*x)) )); // G. C. Greubel, Nov 07 2022

Mathematica

A120730[n_, k_]:= If[n>2*k, 0, Binomial[n,k]*(2*k-n+1)/(k+1)];
A126386[n_]:= Sum[3^k*A120730[n, n-k], {k,0,n}];
Table[A126386[n], {n,0,50}] (* 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 A126386(n): return sum(3^k*A120730(n,n-k) for k in range(n+1))
[A126386(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

Formula

G.f.: (sqrt(1-12*x^2)+2*x-1)/(2*x*(1-4*x)).

a(n) = (1/(n+1))*Sum_{k=0..n+1} Sum_{j=0..k} C(n,k)*C(k,j)*C(2*n-2*k+j, n-2*k+j)*(-1)^(n+j)*2^(2*k-j).

a(n) = Sum_{k=0..floor(n/2)} C(n,n-k)*(n-2*k+1)*3^k/(n-k+1);

a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*3^k.

a(n) = Sum_{k=0..n} 3^k*A120730(n,n-k). - Philippe Deléham, Mar 03 2007

D-finite with recurrence (n+1)*a(n) - 4*(n+1)*a(n-1) + 12*(2-n)*a(n-2) + 48*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 14 2011