A098522 E.g.f. exp(x)*BesselI(2,2*sqrt(3)*x)/3.

0, 0, 1, 3, 18, 70, 330, 1386, 6160, 26496, 115965, 502975, 2194302, 9553050, 41687737, 181908195, 794770200, 3474159304, 15199740171, 66541189473, 291507681070, 1277822445690, 5604712643376, 24596642511628, 108001447419048, 474459925386600, 2085333645995275, 9169506194833881

Offset: 0

Author

Paul Barry, Sep 12 2004

Keywords

• easy
• nonn

Comments

Binomial transform of e.g.f. BesselI(2,2*sqrt(3)x)/3, or {0,0,1,0,12,0,135,0,1512,0,17010,...} with g.f. ((1-6x^2)-sqrt(1-12x^2))/(18x^2*sqrt(1-12x^2)).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A098519, A098521.

Programs

Mathematica

Table[SeriesCoefficient[(1-2*x-5*x^2-(1-x)*Sqrt[1-2*x-11*x^2])/(18*x^2*Sqrt[1-2*x-11*x^2]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)

PARI

x='x+O('x^66); concat([0,0],Vec((1-2*x-5*x^2-(1-x)*sqrt(1-2*x-11*x^2))/(18*x^2*sqrt(1-2*x-11*x^2)))) \\ Joerg Arndt, May 12 2013

Formula

G.f.: (1-2*x-5*x^2-(1-x)*sqrt(1-2*x-11*x^2))/(18*x^2*sqrt(1-2*x-11*x^2)).

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k+2)*3^k.

D-finite with recurrence: (n-2)*(n+2)*a(n) - n*(2n-1)*a(n-1) - 11n*(n-1)*a(n-2) = 0. - R. J. Mathar, Dec 11 2011

a(n) ~ sqrt(18+3*sqrt(3))*(1+2*sqrt(3))^n/(18*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012