%I #26 Sep 08 2022 08:45:15

%S 1,1,4,6,24,40,160,280,1120,2016,8064,14784,59136,109824,439296,

%T 823680,3294720,6223360,24893440,47297536,189190144,361181184,

%U 1444724736,2769055744,11076222976,21300428800,85201715200,164317593600

%N E.g.f. BesselI(0,2*sqrt(2)*x) + BesselI(1,2*sqrt(2)*x)/sqrt(2).

%C Third binomial transform (shifted right) is A047781. Hankel transform is A166232(n+1).

%H G. C. Greubel, <a href="/A098660/b098660.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/sqrt(1-8*x^2)+(1-sqrt(1-8*x^2))/(4*x*sqrt(1-8*x^2)) = (1+4*x-sqrt(1-8*x^2))/(4*x*sqrt(1-8*x^2)).

%F a(n) = binomial(n, floor(n/2))2^floor(n/2).

%F a(n+1) = (1/Pi)*int(x^n*(x+4)/sqrt(8-x^2),x,-2*sqrt(2),2*sqrt(2)) if n is odd [corrected by _Vaclav Kotesovec_, Nov 13 2017].

%F Conjecture: (n+1)*a(n) +(n-1)*a(n-1) -n*a(n-2) +(2-n)*a(n-3) = 0. - _R. J. Mathar_, Nov 15 2011

%t nmax = 30; CoefficientList[Series[BesselI[0, 2*Sqrt[2]*x] + BesselI[1, 2*Sqrt[2]*x]/Sqrt[2], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Nov 13 2017 *)

%o (PARI) x='x+O('x^30); Vec((1+4*x-sqrt(1-8*x^2))/(4*x*sqrt(1-8*x^2))) \\ _G. C. Greubel_, Aug 17 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1+4*x-Sqrt(1-8*x^2))/(4*x*Sqrt(1-8*x^2)))); // _G. C. Greubel_, Aug 17 2018

%Y Cf. A059304, A069720 (bisections).

%K nonn,easy

%O 0,3

%A _Paul Barry_, Sep 20 2004