A098662 Expansion of e.g.f. BesselI(0,2*sqrt(3)*x) + BesselI(1,2*sqrt(3)*x)/sqrt(3).

1, 1, 6, 9, 54, 90, 540, 945, 5670, 10206, 61236, 112266, 673596, 1250964, 7505784, 14073345, 84440070, 159497910, 956987460, 1818276174, 10909657044, 20827527084, 124965162504, 239516561466, 1437099368796, 2763652632300, 16581915793800, 31979409030900, 191876454185400

Offset

0,3

Comments

Fourth binomial transform is A098663.

Links

Robert Israel, Table of n, a(n) for n = 0..1855

Formula

G.f.: 1/sqrt(1-12*x^2) + (1-sqrt(1-12*x^2))/(6*x*sqrt(1-12*x^2));

G.f.: (1 + 6*x - sqrt(1-12*x^2))/(6*x*sqrt(1-12*x^2));

a(n) = binomial(n, floor(n/2))*3^floor(n/2).

Conjecture: (n+1)*a(n) + 6(n-1)*a(n-1) - 12n*a(n-2) + 72*(2-n)*a(n-3) = 0. - R. J. Mathar, Dec 08 2011

Conjecture confirmed using the differential equation x*(6x+1)*(12*x^2-1) * g'(x) + (6*x-1)*(12*x^2+6*x+1)*g(x) + 2*x + 1 = 0 satisfied by the g.f. - Robert Israel, Aug 23 2019

From Amiram Eldar, Nov 16 2025: (Start)

a(n) ~ c * 2^(n+1/2) * 3^(n/2) / sqrt(Pi*n), where c = 1 if n is even, and c = 1/sqrt(3) if n is odd.

Sum_{n>=0} 1/a(n) = 18/11 + 84*arccosec(2*sqrt(3))/(11*sqrt(11)).

Sum_{n>=0} (-1)^n/a(n) = 6/11 - 60*arccosec(2*sqrt(3))/(11*sqrt(11)). (End)

Maple

seq(binomial(n, floor(n/2))*3^floor(n/2), n=0..30); # Robert Israel, Aug 23 2019

Mathematica

With[{nn=30}, CoefficientList[Series[BesselI[0, 2Sqrt[3]x]+ BesselI[1, 2Sqrt[3]x]/ Sqrt[3], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 01 2013 *)

Crossrefs

Cf. A098663.

Sequence in context: A372499 A340819 A254207 * A056425 A056416 A061401

Adjacent sequences: A098659 A098660 A098661 * A098663 A098664 A098665

Keyword

easy,nonn

Author

Paul Barry, Sep 20 2004

Status

approved