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A092145
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Numerator of I(n) = 2*(Integral_{x=0..1/2} (1+x^2)^n dx).
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1
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1, 13, 283, 8667, 342969, 16671885, 962672355, 64467073755, 4917699360945, 421377918441165, 40104072098340075, 4200511400073848475, 480454695780380469225, 59617988532820945752525, 7980059238850231812954675, 1146519564522299271411982875
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OFFSET
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0,2
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COMMENTS
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The denominator is A034910(n+1) = 2^(n-1)*(2*n+2)!/(n+1)!.
The terms in the sequence are numerators of unreduced fractions. They equal the value of the integral multiplied by b(n). The reduced fractions are 1, 13/12, 283/240, 2889/2240, 114323/80640, 1111459/709632, 21392719/12300288 etc. - R. J. Mathar, Nov 24 2008
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LINKS
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FORMULA
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a(n) = (2^(3*n+1)*Gamma(n+3/2)/sqrt(Pi))*Hypergeometric2F1([-n, 1/2], [3/2], -1/4). - Gerry Martens, Aug 09 2015
a(n) = Sum_{k=0..n} binomial(n,k)/(4^k*(2*k+1)). - G. C. Greubel, Feb 05 2024
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EXAMPLE
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I(3) = 8667/6720.
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MAPLE
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f:= n -> simplify(hypergeom([1/2, -n], [3/2], -1/4)*(2*n+2)!*2^(n-1)/(n+1)!):
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MATHEMATICA
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a[n_]:= (2^(1+3*n)*Gamma[3/2+n]*Hypergeometric2F1[-n, 1/2, 3/2, -1/4] )/Sqrt[Pi];
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PROG
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(Magma) [Numerator(&+[Binomial(n, k)/(4^k*(2*k+1)): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 05 2024
(SageMath) [numerator(sum(binomial(n, k)/(4^k*(2*k+1)) for k in range(n+1))) for n in range(31)] # G. C. Greubel, Feb 05 2024
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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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Al Hakanson (hawkuu(AT)excite.com), Mar 31 2004
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EXTENSIONS
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STATUS
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approved
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