login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A278144 Decimal expansion of (sqrt(Pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2. 2
9, 0, 9, 1, 7, 2, 7, 9, 4, 5, 4, 6, 9, 2, 9, 7, 0, 0, 7, 3, 9, 7, 7, 8, 8, 5, 4, 2, 8, 2, 6, 5, 1, 2, 2, 5, 7, 2, 0, 5, 2, 7, 2, 9, 9, 5, 9, 2, 2, 0, 5, 2, 2, 8, 3, 8, 6, 4, 1, 4, 0, 2, 1, 8, 3, 7, 2, 2, 3, 6, 4, 8, 1, 1, 1, 2, 7, 1, 8, 9, 9, 3, 2, 3, 2, 5, 6, 7, 4, 0, 5, 7, 0, 5, 1, 3, 7, 9, 5, 3, 3, 7, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This is the value of hypergeometric([1/4,1/4],[1],-1)^2. See A278143/A241756 for the partial sums of the hypergeometric series hypergeometric([1/2/,1/2,1/2],[1,1],-1) which has this value due to Clausen's formula. See the Hardy reference, p. 106, eq. (7.4.4) where this value is written as (Gamma(9/8)/(Gamma(5/4)*Gamma(7/8)))^2.

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..5000

FORMULA

hypergeometric([1/2/,1/2,1/2],[1,1],-1) = hypergeometric([1/4,1/4],[1],-1)^2 = Sum_{k>=0} (-1)^k*(risefac(k,1/2)/k!)^3, where risefac(x,m) = Product_{j =0..m-1} (x+j), and risefac(x,0) = 1.

(Gamma(9/8)/(Gamma(5/4)*Gamma(7/8)))^2 = (sqrt(Pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2.

EXAMPLE

The value of the series 1 - (1/2)^3 + (1*3/(2*4))^3 - (1*3*5/(2*4*6) + ... is 0.909172794546929700739778854282651225720527299592205228386414021837...

This is also the value of the series Sum_{n>=0} c(n) with c(n) = Sum_{k=0..n} a(k)*a(n-k), where a(0)=1 and a(k) = (-1)^k*(1*5*9 *** (4*k-3)/(4*8*12 *** (4*k)))^2, k >= 1 (self-convolution of the  hypergeometric([1/4,1/4],[1],-1) series).

MATHEMATICA

RealDigits[(Pi/Sqrt[2])*(1/(Gamma[5/8]*Gamma[7/8]))^2, 10, 50][[1]] (* G. C. Greubel, Jan 12 2017 *)

PROG

(PARI) (sqrt(Pi)/(2^(1/4)*gamma(5/8)*gamma(7/8)))^2 \\ Felix Fröhlich, Nov 15 2016

(MAGMA) pi:=Pi(RealField(110)); (Sqrt(pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2 // Felix Fröhlich, Nov 15 2016

CROSSREFS

Cf. A278143.

Sequence in context: A021015 A010680 A248724 * A198220 A131566 A264156

Adjacent sequences:  A278141 A278142 A278143 * A278145 A278146 A278147

KEYWORD

nonn,cons

AUTHOR

Wolfdieter Lang, Nov 14 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 1 03:09 EST 2021. Contains 341732 sequences. (Running on oeis4.)