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A267981
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a(n) = Catalan(n)^2*(4n + 2).
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1
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2, 6, 40, 350, 3528, 38808, 453024, 5521230, 69526600, 898283672, 11848435872, 158966514616, 2163449607200, 29802622140000, 414852500188800, 5827381213589550, 82510878636707400, 1176544010190087000, 16882265852589060000, 243611096252860135800
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OFFSET
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0,1
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COMMENTS
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Numerator of (4n+2)*(Wallis-Lambert-series-1)(n) with denominator A013709(n) convergent to 2*(1-2/Pi). Proof: Both the Wallis-Lambert-series-1=4/Pi-1 and the elliptic Euler-series=1-2/Pi are absolutely convergent series. Thus any linear combination of the terms of these series will be also absolutely convergent to the value of the linear combination of these series - in this case to 2*(1-2/Pi). Q.E.D.
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LINKS
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FORMULA
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Recurrence: (n+1)^2*a(n) = 4*(2*n - 1)*(2*n + 1)*a(n-1). - Vaclav Kotesovec, Jul 16 2016
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EXAMPLE
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For n=3 the a(3)=350.
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MATHEMATICA
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PROG
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(PARI) a000108(n) = binomial(2*n, n)/(n+1)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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