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%I #15 Jun 04 2016 01:47:38
%S 1,3,11,33,99,293,869,2579,7667,22821,68001,202799,605229,1807263,
%T 5399195,16136513,48243347,144275093,431573297,1291258319,3864163769,
%U 11565703931,34622195135,103656406949,310377872861,929465445743
%N Expansion of exp(3*x) - exp(x)*(1-BesselI_0(2*x)).
%C Binomial transform of A081672.
%H G. C. Greubel, <a href="/A081673/b081673.txt">Table of n, a(n) for n = 0..500</a>
%F E.g.f. exp(3*x) - exp(x)*(1-BesselI_0(2*x)).
%F Conjecture: n*(2*n - 7)*a(n) +(-12*n^2 + 50*n - 33)*a(n-1) +(16*n^2 - 76*n + 87)*a(n-2) +3*(4*n^2 - 22*n + 27)*a(n-3) -9*(2*n-5)*(n-3)*a(n-4)=0. - _R. J. Mathar_, Nov 24 2012
%F a(n) ~ 3^n * (1 + sqrt(3)/(2*sqrt(Pi*n))). - _Vaclav Kotesovec_, Jul 02 2015
%F From _Robert Israel_, Jun 03 2016: (Start)
%F E.g.f. A(x) satisfies
%F -27 A + (-63 x + 9) A' + (-18 x^2 + 42 x + 39) A'' + (12 x^2 + 68 x - 25) A''' + (16 x^2 - 58 x + 4) A'''' + (-12 x^2 + 11 x) A''''' + 2 x^2 A'''''' = 0.
%F This implies Mathar's conjectured recursion. (End)
%p Egf:= exp(3*x)-exp(x)*(1-BesselI(0,2*x)):
%p S:= series(Egf,x,101):
%p seq(coeff(S,x,n)*n!, n=0..100); # _Robert Israel_, Jun 03 2016
%t CoefficientList[Series[E^(3*x)-E^x*(1-BesselI[0,2*x]), {x, 0, 50}], x] * Range[0, 50]! (* _Vaclav Kotesovec_, Jul 02 2015 *)
%Y Cf. A025191, A027914, A027915.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 28 2003
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