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

Author

Wolfdieter Lang, Nov 14 2016

Keywords

• nonn
• frac
• easy

Comments

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).

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).

Crossrefs

Cf. A241756, A277232, A278144.

Programs

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.