A098518 E.g.f. exp(x)*BesselI(1,2*sqrt(2)*x)/sqrt(2).

0, 1, 2, 9, 28, 105, 366, 1337, 4824, 17649, 64570, 237545, 875700, 3238105, 11998182, 44550105, 165701168, 617297761, 2302877682, 8602038473, 32168532940, 120425227209, 451253210078, 1692411415161, 6352491269640, 23862066905425, 89696201471786, 337381127856297, 1269781909434724

Offset: 0

Author

Paul Barry, Sep 12 2004

Keywords

• easy
• nonn

Comments

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)).

Links

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

Programs

Mathematica

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

Formula

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