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A277235
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Decimal expansion of 2/(Gamma(3/4))^4.
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3
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8, 8, 6, 9, 4, 1, 1, 6, 8, 5, 7, 8, 1, 1, 5, 4, 0, 5, 4, 1, 1, 5, 2, 5, 3, 6, 1, 3, 5, 4, 5, 2, 1, 5, 3, 8, 6, 8, 6, 4, 9, 9, 9, 1, 9, 6, 4, 2, 5, 9, 8, 3, 4, 8, 3, 0, 9, 8, 6, 0, 9, 8, 9, 8, 1, 3, 1, 7, 8, 2, 5, 5, 9, 4, 8, 1, 9, 2, 7, 9, 7, 0, 6, 9, 1, 5, 2, 6, 4, 7, 7, 9, 4, 9, 8, 1, 2, 1
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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 one of Ramanujan's series: 1 - 5*(1/2)^5 + 9*(1*3/(2*4))^5 -13*(1*3*5/(2*4*6))^5 + - ... . See the Hardy reference p.7. eq. (1.4) and pp. 105-106. For the partial sums see A278140.
The proof of Hardy and Whipple mentioned in the Hardy reference reduces this series to (2/Pi)*Morley's series (for m=1/2). For this series see A277232 and A091670.
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REFERENCES
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G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106, 111.
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LINKS
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FORMULA
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Equals Sum_{k=0..n} (1+4*k)*(binomial(-1/2,k))^5 = Sum_{k=0..n} (-1)^k*(1+4*k)*((2*k-1)!!/(2*k)!!)^5. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
Equals (Gamma(1/4)/Pi)^4/2.
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EXAMPLE
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2/Gamma(3/4)^4 = 0.88694116857811540541152...
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MATHEMATICA
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RealDigits[2/(Gamma[3/4])^4, 10, 100][[1]] (* G. C. Greubel, Oct 26 2018 *)
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PROG
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(Magma) SetDefaultRealField(RealField(100)); 2/(Gamma(3/4))^4; // G. C. Greubel, Oct 26 2018
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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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