0,3

Binomial transform of e.g.f. BesselI(1,2*sqrt(2)*x)/sqrt(2), or {0,1,0,6,0,40,0,280,0,2016,0,....} with g.f. 2*x/(1-8*x^2+sqrt(1-8*x^2)). The binomial transform of e.g.f. BesselI(1,2*sqrt(r)*x)/sqrt(r) with g.f. 2*x/(1-(2*sqrt(r)*x)^2+sqrt(1-(2*sqrt(r)*x)^2)) has g.f. 2*x/(1-2*x-((2*sqrt(r))^2-1)*x^2+(1-x)*sqrt(1-2*x-((2*sqrt(r))^2-1)*x^2)).

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

G.f.: 2*x/(1-2*x-7*x^2+(1-x)*sqrt(1-2*x-7*x^2)).

a(n) = sum{k=0..floor(n/2), binomial(n, k)*binomial(n-k, k+1)*2^k}.

D-finite with recurrence (n+1)*a(n) -3*n*a(n-1) -(5*n+3)*a(n-2) +7*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 14 2011

Shorter recurrence (for n>=3): (n-1)*(n+1)*a(n) = n*(2*n-1)*a(n-1) + 7*(n-1)*n*a(n-2). - Vaclav Kotesovec, Dec 28 2012

a(n) ~ sqrt(4+sqrt(2))*(1+2*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Dec 28 2012

CoefficientList[Series[2*x/(1-2*x-7*x^2+(1-x)*Sqrt[1-2*x-7*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 28 2012 *)

(PARI) x='x+O('x^66); concat([0], Vec(2*x/(1-2*x-7*x^2+(1-x)*sqrt(1-2*x-7*x^2)))) \\ Joerg Arndt, May 11 2013

Sequence in context: A248437 A002532 A360023 * A128239 A307400 A323682

Adjacent sequences: A098515 A098516 A098517 * A098519 A098520 A098521

easy,nonn

Paul Barry, Sep 12 2004

approved