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
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 20 2004
STATUS
approved