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%I #19 May 23 2021 12:34:15
%S 1,1,6,9,54,90,540,945,5670,10206,61236,112266,673596,1250964,7505784,
%T 14073345,84440070,159497910,956987460,1818276174,10909657044,
%U 20827527084,124965162504,239516561466,1437099368796,2763652632300
%N E.g.f. BesselI(0,2*sqrt(3)*x) + BesselI(1,2*sqrt(3)*x)/sqrt(3).
%C Fourth binomial transform is A098663.
%H Robert Israel, <a href="/A098662/b098662.txt">Table of n, a(n) for n = 0..1855</a>
%F G.f.: 1/sqrt(1-12*x^2) + (1-sqrt(1-12*x^2))/(6*x*sqrt(1-12*x^2));
%F G.f.: (1 + 6*x - sqrt(1-12*x^2))/(6*x*sqrt(1-12*x^2));
%F a(n) = binomial(n, floor(n/2))*3^floor(n/2).
%F 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
%F 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
%p seq(binomial(n, floor(n/2))*3^floor(n/2),n=0..30); # _Robert Israel_, Aug 23 2019
%t 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 *)
%K easy,nonn
%O 0,3
%A _Paul Barry_, Sep 20 2004
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