A091670 Decimal expansion of Gamma(1/4)^4/(4*Pi^3).

1, 3, 9, 3, 2, 0, 3, 9, 2, 9, 6, 8, 5, 6, 7, 6, 8, 5, 9, 1, 8, 4, 2, 4, 6, 2, 6, 0, 3, 2, 5, 3, 6, 8, 2, 4, 2, 6, 5, 7, 4, 8, 1, 2, 1, 7, 5, 1, 5, 6, 1, 7, 8, 7, 8, 9, 7, 4, 2, 8, 1, 6, 3, 1, 8, 8, 0, 3, 2, 4, 0, 1, 2, 5, 7, 5, 0, 3, 6, 6, 3, 0, 6, 7, 8, 6, 4, 7, 3, 2, 9, 8, 5, 7, 8, 0, 9, 5, 5, 5, 9, 9

Offset

1,2

Comments

Watson's first triple integral.

This is also the value of F. Morley's series from 1902 Sum_{k=0..n} (risefac(k,1/2)/k!)^3 = hypergeometric([1/2,1/2,1/2],[1,1],1) with the rising factorial risefac(n,x). See A277232, also for the Hardy reference and a MathWorld link. - Wolfdieter Lang, Nov 11 2016

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 324.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 256, 6.1.17 , p. 557, 15.1.26.

M. L. Glasser, I. J. Zucker, Extended Watson integrals for the cubic lattices, Proc. Nat. Acad. Sci., Vol. 74, No. 5 (1977), p. 1800-1801.

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.5.1)

Tito Piezas III, Watson's triple integrals.

Eric Weisstein's World of Mathematics, Watson's Triple Integrals.

I. J. Zucker, 70+years of the Watson integrals, J. Stat. Phys., Vol. 145, No. 3 (2011), pp. 591-612.

Formula

From Joerg Arndt, Nov 27 2010: (Start)

Equals 1/agm(1,sqrt(1/2))^2.

Equals Gamma(1/4)^4 / (4*Pi^3) = Pi / (Gamma(3/4))^4 = hypergeom([1/2,1/2],[1],1/2)^2, see the two Abramowitz - Stegun references. (End)

Equals the square of A175574. Equals A000796/A068465^4. - R. J. Mathar, Jun 17 2016

Equals hypergeom([1/2,1/2,1/2],[1,1],1) - Wolfdieter Lang, Nov 12 2016

Equals Sum_{k>=0} binomial(2*k,k)^3/2^(6*k). - Amiram Eldar, Aug 26 2020

Example

1.39320392968567685918424626032536824265748121751561787897...

Maple

Pi/GAMMA(3/4)^4 ; evalf(%) ; # R. J. Mathar, Jun 17 2016

Mathematica

RealDigits[ N[ Gamma[1/4]^4/(4*Pi^3), 102]][[1]] (* Jean-François Alcover, Nov 12 2012, after Eric W. Weisstein *)

Prog

(PARI) 1/agm(sqrt(1/2), 1)^2 \\ Charles R Greathouse IV, Mar 03 2016
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(1/4)^4/(4*Pi(R)^3); // G. C. Greubel, Oct 26 2018

Crossrefs

Cf. A091671, A091672, A277232, A293238 (inverse).

Sequence in context: A336501 A016674 A264918 * A375503 A201416 A072560

Adjacent sequences: A091667 A091668 A091669 * A091671 A091672 A091673

Keyword

nonn,cons,changed

Author

Eric W. Weisstein, Jan 27 2004

Status

approved