A098265 G.f. : 1/(1-2x-23x^2)^(1/2).

1, 1, 13, 37, 289, 1201, 7741, 38053, 227137, 1207009, 6995053, 38591653, 221446369, 1245188881, 7130897437, 40516456357, 232260610177, 1327920945601, 7627285093069, 43787832627493, 252042452907169, 1451244932278129, 8370001674641917, 48303478743113893, 279083099450496961

Offset: 0

Author

Paul Barry, Aug 31 2004

Keywords

• easy
• nonn

Comments

Central coefficient of (1+x+6x^2)^n.

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, A098264.

Programs

Mathematica

Table[SeriesCoefficient[1/Sqrt[1-2*x-23*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

PARI

x='x+O('x^66); Vec(1/(1-2*x-23*x^2)^(1/2)) \\ Joerg Arndt, May 11 2013

Formula

E.g.f.: exp(x)*BesselI(0, 2*sqrt(6)x).

a(n) = sum{k=0..floor(n/2), binomial(n, k)*binomial(n-k, k)*6^k}.

a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*binomial(2k, k)*6^k}.

n*a(n) +(1-2n)*a(n-1) +23(1-n)*a(n-2)=0. (Recurrence (4) in the Noe paper).- Veka Gesell, Jun 26 2012

a(n) ~ sqrt(72+6*sqrt(6))*(1+2*sqrt(6))^n/(12*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012