A129147 Expansion of c(x(1+2x)), c(x) the g.f. of A000108.

1, 1, 4, 13, 52, 214, 928, 4141, 18940, 88258, 417616, 2001058, 9690184, 47348812, 233158144, 1155900541, 5764510060, 28898899594, 145556001136, 736206912982, 3737768204344, 19042072755124, 97313398530496, 498737257238482, 2562773039735896, 13200732624526804, 68148459129343648

Offset: 0

Author

Paul Barry, Apr 01 2007

Keywords

• easy
• nonn

Comments

Hankel transform of a(n) is A047656(n+1)=3^C(n+1,2). In general, the Hankel transform of the expansion of c(x(1+r*x)) is (r+1)^C(n+1,2).

Number of paths weakly above X-axis from (0,0) to (0,2n) using steps (1,1), (1,-1) and two colors of (3,1). - David Scambler, Jun 21 2013

References

• Barry, Paul; Hennessy, Aoife Four-term recurrences, orthogonal polynomials and Riordan arrays. J. Integer Seq. 15 (2012), no. 4, Article 12.4.2, 19 pp.

Links

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

Programs

Mathematica

CoefficientList[Series[(1-Sqrt[1-4*x*(1+2*x)])/(2*x*(1+2*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

PARI

x='x+O('x^66);
C(x)=(1-sqrt(1-4*x))/(2*x);
Vec(C(x*(1+2*x))) \\ Joerg Arndt, May 15 2013

Formula

a(n)=sum{k=0..n, C(k,n-k)*2^(n-k)*C(k)};

a(n)=(1/(2*pi))*int(x^n*sqrt(8+4x-x^2)/(x+2),x,2-2*sqrt(3),2+2*sqrt(3));

Conjecture: (n+1)*a(n) +2*(2-n)*a(n-1) +4*(5-4n)*a(n-2) +16*(2-n)*a(n-3)=0. - R. J. Mathar, Dec 14 2011

G.f.: Q(0), where Q(k)= 1 + (4*k+1)*x*(1+2*x)/(k + 1 - x*(1+2*x)*(2*k+2)*(4*k+3)/(2*x*(1+2*x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013

a(n) ~ sqrt(3-sqrt(3)) * (2*(1+sqrt(3)))^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 01 2014