A098660 E.g.f. BesselI(0,2*sqrt(2)*x) + BesselI(1,2*sqrt(2)*x)/sqrt(2).

1, 1, 4, 6, 24, 40, 160, 280, 1120, 2016, 8064, 14784, 59136, 109824, 439296, 823680, 3294720, 6223360, 24893440, 47297536, 189190144, 361181184, 1444724736, 2769055744, 11076222976, 21300428800, 85201715200, 164317593600

Offset: 0

Author

Paul Barry, Sep 20 2004

Keywords

• nonn
• easy

Comments

Third binomial transform (shifted right) is A047781. Hankel transform is A166232(n+1).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A059304, A069720 (bisections).

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1+4*x-Sqrt(1-8*x^2))/(4*x*Sqrt(1-8*x^2)))); // G. C. Greubel, Aug 17 2018

Mathematica

nmax = 30; CoefficientList[Series[BesselI[0, 2*Sqrt[2]*x] + BesselI[1, 2*Sqrt[2]*x]/Sqrt[2], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 13 2017 *)

PARI

x='x+O('x^30); Vec((1+4*x-sqrt(1-8*x^2))/(4*x*sqrt(1-8*x^2))) \\ G. C. Greubel, Aug 17 2018

Formula

G.f.: 1/sqrt(1-8*x^2)+(1-sqrt(1-8*x^2))/(4*x*sqrt(1-8*x^2)) = (1+4*x-sqrt(1-8*x^2))/(4*x*sqrt(1-8*x^2)).

a(n) = binomial(n, floor(n/2))2^floor(n/2).

a(n+1) = (1/Pi)*int(x^n*(x+4)/sqrt(8-x^2),x,-2*sqrt(2),2*sqrt(2)) if n is odd [corrected by Vaclav Kotesovec, Nov 13 2017].

Conjecture: (n+1)*a(n) +(n-1)*a(n-1) -n*a(n-2) +(2-n)*a(n-3) = 0. - R. J. Mathar, Nov 15 2011