A278143 Numerators of partial sums of a hypergeometric series with value Pi/(sqrt(2)*(Gamma(5/8)*Gamma(7/8))^2) = A278144.
1, 7, 475, 3675, 1924475, 15145753, 981654583, 7774283075, 32109931838075, 255083626080725, 16423892777415669, 130705503226766013, 67230186897380845975, 535644114907108845925, 34407319668610517498575, 274347338677567001587475
Offset: 0
Examples
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.
References
- G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 106, eq. (7.4.4).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..500
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 557, (15.1.21).
Programs
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Mathematica
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 *) -
PARI
for(n=0,25, print1(numerator(sum(k=0,n, binomial(-1/2,k)^3)), ", ")) \\ G. C. Greubel, Feb 06 2017
Formula
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.
Comments