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A081672 Expansion of exp(2x) - exp(0) + BesselI_0(2x).

%I A081672 #28 Jun 14 2016 23:03:28
%S A081672 1,2,6,8,22,32,84,128,326,512,1276,2048,5020,8192,19816,32768,78406,
%T A081672 131072,310764,524288,1233332,2097152,4899736,8388608,19481372,
%U A081672 33554432,77509464,134217728,308552056,536870912,1228859344
%N A081672 Expansion of exp(2x) - exp(0) + BesselI_0(2x).
%C A081672 Inverse binomial transform of A081673.
%H A081672 G. C. Greubel, <a href="/A081672/b081672.txt">Table of n, a(n) for n = 0..500</a>
%F A081672 E.g.f.: exp(2x) - exp(0) + BesselI_0(2x).
%F A081672 Conjecture: n*a(n) +2*(1-n)*a(n-1) +4*(1-n)*a(n-2) +8*(n-2)*a(n-3)=0. - _R. J. Mathar_, Nov 12 2012
%F A081672 a(n) ~ 2^n * (1+(1+(-1)^n)/sqrt(2*Pi*n)). - _Vaclav Kotesovec_, Feb 04 2014
%F A081672 From _Benedict W. J. Irwin_, Jun 03 2016: (Start)
%F A081672 For odd n, a(n) = 2^n. For even n>0, a(n) = 2^n*(1+n!/(2^n*(n/2)!^2)).
%F A081672 G.f.: 1/sqrt(1-4*z^2) + 1/(1-2*z) - 1. (End)
%F A081672 E.g.f. satisfies y''' - (2*x-2)*y'' - (4*x + 2)*y' + (8*x-4)*y + 8x - 4 = 0, which implies Mathar's conjectured recurrence. - _Robert Israel_, Jun 03 2016
%p A081672 1, seq(op([2^(2*k-1), 2^(2*k)+(2*k)!/k!^2]), k=1..30); # _Robert Israel_, Jun 03 2016
%t A081672 CoefficientList[Series[1/Sqrt[1 - 4 z^2] + 1/(1 - 2 z) - 1, {z, 0, 20}], z] (* _Benedict W. J. Irwin_, Jun 03 2016 *)
%t A081672 CoefficientList[Series[Exp[2*x] - 1 + BesselI[0, 2*x], {x, 0, 50}],
%t A081672   x]*Range[0, 50]! (* _G. C. Greubel_, Jun 03 2016 *)
%o A081672 (PARI) a(n)=if(n,if(n%2,1,1+n!/(2^n*(n/2)!^2))<<n,1) \\ _Charles R Greathouse IV_, Jun 10 2016
%o A081672 (PARI) Vec(1/sqrt(1-4*x^2)+1/(1-2*x)-1) \\ _Charles R Greathouse IV_, Jun 10 2016
%Y A081672 Cf. A000984, A081673.
%K A081672 easy,nonn
%O A081672 0,2
%A A081672 _Paul Barry_, Mar 28 2003