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a(n) = Integral_{x>=0} 2*BesselK(0, 2*sqrt(x))*x^n. This is an integral representation as represents the n-th moment of a positive function defined on the positive half-axis. - Karol A. Penson, Oct 09 2001
Sum_{n>=0} a(n) * x^= (2*n+1)/! * [x^(2*n+1)! = ] 4*arcsin(x/2)/sqrt(4-x^2). - Ira M. Gessel, Dec 10 2024
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a(n) = Integral _{x>=0} 2*BesselK(0, 2*sqrt(x))*x^n. This is an integral representation as the n-th moment of a positive function on a the positive half-axis, in Maple notation: a(n)=int(x^n*2*BesselK(0, 2*sqrt(x)), x=0..infinity), n=0, 1... - Karol A. Penson, Oct 09 2001
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A(x) = Sum_{n>=0,N) a(n)*x^n = 1 + x/(U(0;N-2)-x); N >= 4; U(k)= 1 + x*(k+1)^2 - x*(k+2)^2/G(k+1); besides U(0;infinity)=x; (continued fraction, Euler's 1st kind, 1-step).
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Sum_{n>=0} a(n) * x^(2*n+1)/(2*n+1)! = 4*arcsin(x/2)/sqrt(4-x^2). - Ira M. Gessel, Dec 10 2024
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