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A278144
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Decimal expansion of (sqrt(Pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2.
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2
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9, 0, 9, 1, 7, 2, 7, 9, 4, 5, 4, 6, 9, 2, 9, 7, 0, 0, 7, 3, 9, 7, 7, 8, 8, 5, 4, 2, 8, 2, 6, 5, 1, 2, 2, 5, 7, 2, 0, 5, 2, 7, 2, 9, 9, 5, 9, 2, 2, 0, 5, 2, 2, 8, 3, 8, 6, 4, 1, 4, 0, 2, 1, 8, 3, 7, 2, 2, 3, 6, 4, 8, 1, 1, 1, 2, 7, 1, 8, 9, 9, 3, 2, 3, 2, 5, 6, 7, 4, 0, 5, 7, 0, 5, 1, 3, 7, 9, 5, 3, 3, 7, 3
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OFFSET
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0,1
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COMMENTS
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This is the value of hypergeometric([1/4,1/4],[1],-1)^2. See A278143/A241756 for the partial sums of the hypergeometric series hypergeometric([1/2/,1/2,1/2],[1,1],-1) which has this value due to Clausen's formula. See the Hardy reference, p. 106, eq. (7.4.4) where this value is written as (Gamma(9/8)/(Gamma(5/4)*Gamma(7/8)))^2.
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REFERENCES
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G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 106, eq. (7.4.4)
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LINKS
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FORMULA
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Equals hypergeometric([1/2/,1/2,1/2],[1,1],-1) = hypergeometric([1/4,1/4],[1],-1)^2 = Sum_{k>=0} (-1)^k*(risefac(k,1/2)/k!)^3, where risefac(x,m) = Product_{j =0..m-1} (x+j), and risefac(x,0) = 1.
Equals (Gamma(9/8)/(Gamma(5/4)*Gamma(7/8)))^2 = (sqrt(Pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2.
Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)^3/64^k. - Amiram Eldar, Jul 04 2023
Equals Gamma(1/8)^4 * (2 - sqrt(2)) / (16 * Pi^2 * Gamma(1/4)^2). - Vaclav Kotesovec, Jul 04 2023
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EXAMPLE
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The value of the series 1 - (1/2)^3 + (1*3/(2*4))^3 - (1*3*5/(2*4*6) + ... is 0.909172794546929700739778854282651225720527299592205228386414021837...
This is also the value of the series Sum_{n>=0} c(n) with c(n) = Sum_{k=0..n} f(k)*f(n-k), where f(0)=1 and f(k) = (-1)^k*(1*5*9 *** (4*k-3)/(4*8*12 *** (4*k)))^2, k >= 1 (self-convolution of the hypergeometric([1/4,1/4],[1],-1) series).
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MATHEMATICA
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RealDigits[(Pi/Sqrt[2])*(1/(Gamma[5/8]*Gamma[7/8]))^2, 10, 50][[1]] (* G. C. Greubel, Jan 12 2017 *)
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PROG
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(PARI) (sqrt(Pi)/(2^(1/4)*gamma(5/8)*gamma(7/8)))^2 \\ Felix Fröhlich, Nov 15 2016
(Magma) pi:=Pi(RealField(110)); (Sqrt(pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2 // Felix Fröhlich, Nov 15 2016
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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