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

0,3

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

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

From Amiram Eldar, Dec 04 2025: (Start)

Sum_{n>=0} 1/a(n) = 12/7 + 40*sqrt(7)*arccot(sqrt(7))/49.

Sum_{n>=0} (-1)^n/a(n) = 4/7 - 24*sqrt(7)*arccosec(2*sqrt(2))/49. (End)

From Vaclav Kotesovec, Dec 04 2025: (Start)

Sum_{n>=0} 1/a(n) = 12/7 + 40*arcsin(2^(-3/2))/7^(3/2).

Sum_{n>=0} (-1)^n/a(n) = 4/7 - 24*arcsin(2^(-3/2))/7^(3/2). (End)

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 *)

Prog

(PARI) my(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
(Magma) m:=30; R<x>:=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

Crossrefs

Cf. A047781, A059304, A069720 (bisections), A166232.

Sequence in context: A071224 A305381 A164532 * A363631 A122174 A283185

Adjacent sequences: A098657 A098658 A098659 * A098661 A098662 A098663

Keyword

nonn,easy

Author

Paul Barry, Sep 20 2004

Status

approved