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%I #19 May 23 2021 02:41:55
%S 0,1,2,12,40,185,726,3157,13112,56331,239230,1028522,4414224,19045039,
%T 82237442,356104140,1544056864,6707220443,29172892518,127058629852,
%U 554006070200,2418204764451,10565384173762,46202462762837,202207635999240,885642000534925,3881697614968706
%N E.g.f. exp(x)*BesselI(1,2*sqrt(3)*x)/sqrt(3).
%C Binomial transform of e.g.f. BesselI(1,2*sqrt(3)x)/sqrt(3), or {0,1,0,9,0,90,0,945,0,10206,0,...} with g.f. 2x/(1-12x^2+sqrt(1-12x^2)). The binomial transform of e.g.f. BesselI(1,2*sqrt(r)x)/sqrt(r) with g.f. 2x/(1-(2*sqrt(r)x)^2+sqrt(1-(2*sqrt(r)x)^2)) has g.f. 2x/(1-2x-((2*sqrt(r))^2-1)x^2+(1-x)*sqrt(1-2x-((2*sqrt(r))^2-1)x^2)).
%H Vincenzo Librandi, <a href="/A098519/b098519.txt">Table of n, a(n) for n = 0..300</a>
%F G.f.: 2*x/(1-2*x-11*x^2+(1-x)*sqrt(1-2*x-11*x^2)).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k+1)*3^k.
%F Conjecture: (n-1)*(n+1)*a(n) - n*(2*n-1)*a(n-1) - 11*n*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 23 2011
%F a(n) ~ sqrt(6+sqrt(3))*(1+2*sqrt(3))^n/(6*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 15 2012
%t Table[SeriesCoefficient[2*x/(1-2*x-11*x^2+(1-x)*Sqrt[1-2*x-11*x^2]),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 15 2012 *)
%t With[{nn=30},CoefficientList[Series[Exp[x] BesselI[1,2x Sqrt[3]]/Sqrt[3],{x,0,nn}],x] Range[0,nn]!]//Simplify (* _Harvey P. Dale_, Apr 27 2016 *)
%o (PARI) x='x+O('x^66); concat([0],Vec(2*x/(1-2*x-11*x^2+(1-x)*sqrt(1-2*x-11*x^2)))) \\ _Joerg Arndt_, May 11 2013
%K easy,nonn
%O 0,3
%A _Paul Barry_, Sep 12 2004
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