A278141 - OEIS

%I #21 Jan 10 2017 08:06:32

%S 1,265,1096065,281858265,18519577975665,4748934018906441,

%T 19474365987782658225,4989739877102195271225,

%U 5235591401647346852339166225,1341015791319444602368386319225,5495144390631448939048252704196225,1407253983507773608409169421000239225,92253220393640211712365553562313715740225

%N Numerators of partial sums of a Ramanujan series converging to 2^(3/2)/(sqrt(Pi)*Gamma(3/4)^2) given in A278146.

%C The denominators are given in A278142.

%C 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.

%C 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)).

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

%H G. C. Greubel, <a href="/A278141/b278141.txt">Table of n, a(n) for n = 0..100</a>

%F 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.

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

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

%e The value of the series is (see A278143)

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

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

%o (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

%Y Cf. A278142, A278146.

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 14 2016