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A278141 Numerators of partial sums of a Ramanujan series converging to 2^(3/2)/(sqrt(Pi)*Gamma(3/4)^2) given in A278146. 3
1, 265, 1096065, 281858265, 18519577975665, 4748934018906441, 19474365987782658225, 4989739877102195271225, 5235591401647346852339166225, 1341015791319444602368386319225, 5495144390631448939048252704196225, 1407253983507773608409169421000239225, 92253220393640211712365553562313715740225 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The denominators are given in A278142.

One of Ramanujan's series is  1 + 9*(1/4)^4 + 17*(1*5/(4*8))^4 + 25*(1*5*9/(4*8*12))^4 + ... = Sum_{k>=0} (1+8*k)*(risefac(1/4,k)/k!)^4 where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, p. 7, eq. (1.3) and p. 105, eq. (7.4.3) for s=1/4. The value of this series is 2^(3/2)/(sqrt(Pi)*Gamma(3/4)^2) given in A278146.

The general formula, Hardy, p. 105, eq. (7.4.3) (divided by s) is Sum_{k>=0} (1 + 2*k/s)*(risefac(s,k)/k!)^4 = sin^2(s*Pi)*Gamma(s)^2/(2*s*Pi^2*cos(s*Pi)* Gamma(2*s)).

REFERENCES

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..100

FORMULA

a(n) = numerator(r(n)), with the rationals r(n) = Sum_{k=0..n} (1+8*k)*(risefac(1/4,k)/k!)^4. The rising factorial has been defined in a comment above.

a(n) = Sum_{k=0..n} (1+8*k)*(binomial(-1/4,k))^4.

EXAMPLE

The rationals begin: 1, 265/256, 1096065/1048576, 281858265/268435456, 18519577975665/17592186044416, 4748934018906441/4503599627370496, 19474365987782658225/18446744073709551616, ...

The value of the series is (see A278143)

  2^(3/2)/(sqrt(Pi)*Gamma(3/4)^2) = 1.06267989991... .

MATHEMATICA

Numerator[Table[ Sum[  (1 + 8*k)*(Binomial[-1/4, k])^4 , {k, 0, n}] , {n, 0, 25}]] (* G. C. Greubel, Jan 09 2017 *)

PROG

(PARI) for(n=0, 10, print1( numerator( sum(k=0, n, (1+8*k)*(binomial(-1/4, k))^4)), ", ")) \\ G. C. Greubel, Jan 09 2017

CROSSREFS

Cf. A278142, A278146.

Sequence in context: A211719 A210120 A266308 * A091676 A061662 A321224

Adjacent sequences:  A278138 A278139 A278140 * A278142 A278143 A278144

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Nov 14 2016

STATUS

approved

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Last modified April 18 16:51 EDT 2021. Contains 343089 sequences. (Running on oeis4.)