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
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
MATHEMATICA
A120730[n_, k_]:= If[n>2*k, 0, Binomial[n, k]*(2*k-n+1)/(k+1)];
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)
[A126386(n) for n in range(51)] # G. C. Greubel, Nov 07 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 28 2007
STATUS
approved