A098522 - OEIS

%I #22 Feb 14 2024 07:48:51

%S 0,0,1,3,18,70,330,1386,6160,26496,115965,502975,2194302,9553050,

%T 41687737,181908195,794770200,3474159304,15199740171,66541189473,

%U 291507681070,1277822445690,5604712643376,24596642511628,108001447419048,474459925386600,2085333645995275,9169506194833881

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

%C 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)).

%H Vincenzo Librandi, <a href="/A098522/b098522.txt">Table of n, a(n) for n = 0..300</a>

%F 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)).

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

%F 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

%F a(n) ~ sqrt(18+3*sqrt(3))*(1+2*sqrt(3))^n/(18*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 15 2012

%t 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 *)

%o (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

%Y Cf. A098519, A098521.

%K easy,nonn

%O 0,4

%A _Paul Barry_, Sep 12 2004