OFFSET
0,3
COMMENTS
Binomial transform of e.g.f. BesselI(1,4x)/2, or {0,1,0,12,0,160,0,2240,0,32256,0,...} with g.f. 2x/(1-16x^2+sqrt(1-16x^2)). The binomial transform of e.g.f. BesselI(1,2*sqrt(r)x)/sqrt(r) with g.f. 2x/(1-(2*sqrt(r)x)^2+sqrt(1-(2*sqrt(r)x)^2)) has g.f. 2x/(1-2x-((2*sqrt(r))^2-1)x^2+(1-x)*sqrt(1-2x-((2*sqrt(r))^2-1)x^2)).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
FORMULA
G.f.: 2*x/(1-2*x-15*x^2+(1-x)*sqrt(1-2*x-15*x^2)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k+1)4^k.
Conjecture: (n-1)*(n+1)*a(n) - n*(2n-1)*a(n-1) - 15*n*(n-1)*a(n-2) = 0. - R. J. Mathar, Dec 11 2011
a(n) ~ 5^(n+1/2)/(4*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 15 2012
a(n) = 2^(n-1)*GegenbauerC(n-1, -n, -1/4). - Peter Luschny, May 08 2016
MAPLE
a := n -> simplify(2^(n-1)*GegenbauerC(n-1, -n, -1/4)):
seq(a(n), n=0..26); # Peter Luschny, May 08 2016
MATHEMATICA
Table[SeriesCoefficient[2*x/(1-2*x-15*x^2+(1-x)*Sqrt[1-2*x-15*x^2]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 15 2012 *)
PROG
(PARI) x='x+O('x^66); concat([0], Vec(2*x/(1-2*x-15*x^2+(1-x)*sqrt(1-2*x-15*x^2)))) \\ Joerg Arndt, May 11 2013
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 12 2004
STATUS
approved