A098264 G.f.: 1/(1-2x-19x^2)^(1/2).

1, 1, 11, 31, 211, 851, 4901, 22961, 124531, 623011, 3313201, 17086301, 90453661, 473616781, 2509264811, 13250049551, 70368250451, 373539254611, 1989045489281, 10597110956861, 56566637447401, 302196871378601, 1616570627763311, 8654955238504531, 46384344189261661

Offset: 0

Author

Paul Barry, Aug 31 2004

Keywords

• easy
• nonn

Comments

Central coefficient of (1+x+5x^2)^n.

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have five colors. - N-E. Fahssi, Mar 31 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Crossrefs

Cf. A084601, A084603, A084605.

Programs

Mathematica

Table[SeriesCoefficient[1/Sqrt[1-2*x-19*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

PARI

x='x+O('x^66); Vec(1/(1-2*x-19*x^2)^(1/2)) \\ Joerg Arndt, May 11 2013

Formula

E.g.f. : exp(x)*BesselI(0, 2*sqrt(5)x).

a(n) = sum{k=0..floor(n/2), binomial(n, k)*binomial(n-k, k)*5^k}.

a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*binomial(2k, k)*5^k}.

n*a(n) +(1-2*n)*a(n-1) +19*(1-n)*a(n-2)=0. - R. J. Mathar, Nov 14 2011

a(n) ~ sqrt(50+5*sqrt(5))*(1+2*sqrt(5))^n/(10*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012