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%I #39 Apr 23 2023 04:21:49
%S 1,5,1,6,3,8,6,0,5,9,1,5,1,9,7,8,0,1,8,1,5,6,0,1,2,1,5,9,6,8,1,4,2,0,
%T 7,7,9,9,5,5,3,8,7,0,4,4,4,5,2,2,6,2,6,7,6,5,6,6,9,8,0,4,6,3,6,5,8,0,
%U 8,6,3,2,0,3,5,3,5,2,1,4,5,0,4,0,1,6,1,1,7,4,1,2,0,9,6,8,8,1,1,3,9,2
%N Decimal expansion of value of Watson's integral.
%H G. C. Greubel, <a href="/A086231/b086231.txt">Table of n, a(n) for n = 1..10000</a>
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 40.
%H M. Lawrence Glasser and I. John Zucker, <a href="https://doi.org/10.1073/pnas.74.5.1800">Extended Watson integrals for the cubic lattices</a>, Proceedings of the National Academy of Sciences, Vol. 74, No. 5 (1977), pp. 1800-1801, <a href="https://www.researchgate.net/publication/7188519_Extended_Watson_integrals_for_the_cubic_lattices">alternative link</a>.
%H Anthony J. Guttmann, <a href="http://dx.doi.org/10.1088/1751-8113/43/30/305205">Lattice Green's functions in all dimensions</a>, J. Phys. A.: Math. Theor., Vol. 43, No. 30 (2010) 305205.
%H George N. Watson, <a href="https://doi.org/10.1093/qmath/os-10.1.266">Three triple integrals</a>, The Quarterly Journal of Mathematics, Vol. os-10, No. 1 (1939), pp. 266-276.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolyasRandomWalkConstants.html">Pólya's Random Walk Constants</a>.
%F Equals (sqrt(3)-1)*(gamma(1/24)*gamma(11/24))^2/(32*Pi^3). - _G. C. Greubel_, Jan 07 2018
%F Equals 1/(1 - A086230). - _Amiram Eldar_, Aug 28 2020
%F Equals Sum_{k>=0} A002896(k)/36^k. - _Vaclav Kotesovec_, Apr 23 2023
%e 1.51638605915197801815601215968142077995538704445226267656698...
%p evalf((sqrt(3)-1)*(GAMMA(1/24)*GAMMA(11/24))^2 / (32*Pi^3),120); # _Vaclav Kotesovec_, Sep 16 2014
%t RealDigits[ N[ Sqrt[6]/32/Pi^3*Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24], 102]][[1]] (* _Jean-François Alcover_, Nov 12 2012, after _Eric W. Weisstein_ *)
%o (PARI) (sqrt(3)-1)*(gamma(1/24)*gamma(11/24))^2 / (32*Pi^3) \\ _Altug Alkan_, Apr 13 2016
%o (Magma) C<i> := ComplexField(); (Sqrt(3)-1)*(Gamma(1/24)*Gamma(11/24))^2/(32*Pi(C)^3) // _G. C. Greubel_, Jan 07 2018
%Y Cf. A002896, A086230, A242812, A242813, A242814, A242815, A242816, A273086.
%K nonn,cons
%O 1,2
%A _Eric W. Weisstein_, Jul 12 2003
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