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

0,3

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.

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

Conjecture: (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

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 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (Sqrt(1-12*x^2)+2*x-1)/(2*x*(1-4*x)) )); // 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

Crossrefs

Cf. A000108, A009766, A047656, A120730.

Sequence in context: A061668 A239025 A339393 * A149074 A149075 A149076

Adjacent sequences: A128383 A128384 A128385 * A128387 A128388 A128389

Keyword

easy,nonn

Author

Paul Barry, Feb 28 2007

Status

approved