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A278143
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Numerators of partial sums of a hypergeometric series with value Pi/(sqrt(2)*(Gamma(5/8)*Gamma(7/8))^2) = A278144.
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4
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1, 7, 475, 3675, 1924475, 15145753, 981654583, 7774283075, 32109931838075, 255083626080725, 16423892777415669, 130705503226766013, 67230186897380845975, 535644114907108845925, 34407319668610517498575, 274347338677567001587475
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OFFSET
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0,2
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COMMENTS
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The denominators appear to be given in A241756.
The series is 1 - (1/2)^3 + (1*3/2*4)^3 -+ ... = Sum_{k>=0} (-1)^k*(risefac(1/2,k)/ k!)^3 = hypergeometric([1/2,1/2,1/2],[1,1],-1), where risefac(x,k) = Product_{j =0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference p. 106.
Due to Clausen's formula given in eq. (7.4.5) this is (hypergeometric([1/2,1/2],[1],-1))^2. Hardy's result in eq. (7.4.4) is Gamma(9/8)/(Gamma(5/4)* Gamma(7/8)))^2 which can be rewritten as (sqrt(Pi)/(2^(1/4)*Gamma(5/8)* Gamma(7/8)))^2. See the Abramowitz-Stegun reference p. 557, 15.1.21 and p. 256, 6.1.18.
This series is the alternating sum version of Morley's series for m=1/2. See A277232. Hence the present sequence gives the numerators of the partial sums of the cubes of the expansion coefficients of 1/sqrt(1+x).
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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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a(n) = numerator(r(n)) with the rational r(n) = Sum_{k=0..n} (-1)^k (risefac(1/2,k)/k!)^3 = Sum_{k=0..n} (binomial(-1/2,k))^3 = Sum_{k=0..n} (-1)^k*((2*k-1)!!/(2*k)!!)^3. The rising factorial has been defined in a comment above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
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EXAMPLE
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The rationals begin: 1, 7/8, 475/512, 3675/4096, 1924475/2097152, 15145753/16777216, 981654583/1073741824, 7774283075/8589934592, ... .
The limit r(n), for n -> oo is Pi/(sqrt(2)*(Gamma(5/8)*Gamma(7/8))^2) = 0.90917563087572... given in A278144.
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MATHEMATICA
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Table[Numerator@ Sum[(-1)^k (Pochhammer[1/2, k]/k!)^3, {k, 0, n}], {n, 0, 15}] (* or *)
Table[Numerator@ Sum[Binomial[-1/2, k]^3, {k, 0, n}], {n, 0, 15}] (* or *)
Table[Numerator@ Sum[(-1)^k*((2 k - 1)!!/(2 k)!!)^3, {k, 0, n}], {n, 0, 15}] (* Michael De Vlieger, Nov 15 2016 *)
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PROG
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(PARI) for(n=0, 25, print1(numerator(sum(k=0, n, binomial(-1/2, k)^3)), ", ")) \\ G. C. Greubel, Feb 06 2017
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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