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A119976 E.g.f. exp(2x)*(Bessel_I(0,2*sqrt(2)x) + Bessel_I(1,2*sqrt(2)x)/sqrt(2)).

%I A119976 #24 Sep 08 2022 08:45:25
%S A119976 1,3,12,50,216,952,4256,19224,87520,400928,1845888,8533824,39590656,
%T A119976 184216320,859354112,4017738112,18820855296,88317817344,415075665920,
%U A119976 1953473141760,9205135036416,43425512132608,205072796270592
%N A119976 E.g.f. exp(2x)*(Bessel_I(0,2*sqrt(2)x) + Bessel_I(1,2*sqrt(2)x)/sqrt(2)).
%C A119976 Binomial transform of A119975. Binomial transform is A047781(n+1).
%H A119976 Vincenzo Librandi, <a href="/A119976/b119976.txt">Table of n, a(n) for n = 0..300</a>
%F A119976 G.f.: (1+2*x)/(4*x*sqrt(1-4*x-4*x^2))-1/(4*x);
%F A119976 a(n) = Sum_{k=0..n} 2^(n-k)*C(n,k)*C(k,floor(k/2))2^floor(k/2).
%F A119976 D-finite with recurrence: (n+1)*a(n) -2*(n+2)*a(n-1) +12*(1-n)*a(n-2) +8*(2-n)*a(n-3) = 0. - _R. J. Mathar_, Dec 10 2011
%F A119976 Shorter recurrence: n*(n+1)*a(n) = 2*n*(2*n+1)*a(n-1) + 4*(n-1)*(n+1)*a(n-2). - _Vaclav Kotesovec_, Oct 19 2012
%F A119976 a(n) ~ sqrt(20+14*sqrt(2))*(2+2*sqrt(2))^n/(4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 19 2012
%t A119976 CoefficientList[Series[(1+2*x)/(4*x*Sqrt[1-4*x-4*x^2])-1/(4*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%o A119976 (PARI) x='x+O('x^50); Vec((1+2*x)/(4*x*sqrt(1-4*x-4*x^2))-1/(4*x)) \\ _G. C. Greubel_, Feb 08 2017
%o A119976 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1+2*x)/(4*x*Sqrt(1-4*x-4*x^2)) -1/(4*x))); // _G. C. Greubel_, Aug 17 2018
%K A119976 easy,nonn
%O A119976 0,2
%A A119976 _Paul Barry_, Jun 02 2006