This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A091670 #70 Aug 19 2025 09:33:27
%S A091670 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,
%T A091670 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,
%U A091670 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
%N A091670 Decimal expansion of Gamma(1/4)^4/(4*Pi^3).
%C A091670 Watson's first triple integral.
%C A091670 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
%C A091670 This constant is transcendental due to a result of Nesterenko, who proves that Gamma(1/4) is algebraically independent of Pi. - _Charles R Greathouse IV_, Aug 19 2025
%D A091670 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 324.
%H A091670 G. C. Greubel, <a href="/A091670/b091670.txt">Table of n, a(n) for n = 1..10000</a>
%H A091670 A. M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 256, 6.1.17 , p. 557, 15.1.26.
%H A091670 M. L. Glasser, I. J. Zucker, <a href="https://doi.org/10.1073/pnas.74.5.1800">Extended Watson integrals for the cubic lattices</a>, Proc. Nat. Acad. Sci., Vol. 74, No. 5 (1977), p. 1800-1801.
%H A091670 Paul J. Nahin, <a href="https://doi.org/10.1007/978-3-030-43788-6">Inside interesting integrals</a>, Undergrad. Lecture Notes in Physics, Springer (2020), (6.5.1)
%H A091670 Yu. V. Nesterenko, <a href="https://doi.org/10.1070/SM1996v187n09ABEH000158">Modular functions and transcendence questions</a>, Sbornik: Mathematics, Vol. 187, No. 9 (1996), pp. 1319-1348. (English translation)
%H A091670 Tito Piezas III, <a href="https://sites.google.com/view/tpiezas/0025-part-4-watsons-triple-integrals">Watson's triple integrals</a>.
%H A091670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WatsonsTripleIntegrals.html">Watson's Triple Integrals</a>.
%H A091670 I. J. Zucker, <a href="https://doi.org/10.1007/s10955-011-0273-0">70+years of the Watson integrals</a>, J. Stat. Phys., Vol. 145, No. 3 (2011), pp. 591-612.
%H A091670 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A091670 From _Joerg Arndt_, Nov 27 2010: (Start)
%F A091670 Equals 1/agm(1,sqrt(1/2))^2.
%F A091670 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)
%F A091670 Equals the square of A175574. Equals A000796/A068465^4. - _R. J. Mathar_, Jun 17 2016
%F A091670 Equals hypergeom([1/2,1/2,1/2],[1,1],1) - _Wolfdieter Lang_, Nov 12 2016
%F A091670 Equals Sum_{k>=0} binomial(2*k,k)^3/2^(6*k). - _Amiram Eldar_, Aug 26 2020
%e A091670 1.39320392968567685918424626032536824265748121751561787897...
%p A091670 Pi/GAMMA(3/4)^4 ; evalf(%) ; # _R. J. Mathar_, Jun 17 2016
%t A091670 RealDigits[ N[ Gamma[1/4]^4/(4*Pi^3), 102]][[1]] (* _Jean-François Alcover_, Nov 12 2012, after _Eric W. Weisstein_ *)
%o A091670 (PARI) 1/agm(sqrt(1/2),1)^2 \\ _Charles R Greathouse IV_, Mar 03 2016
%o A091670 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(1/4)^4/(4*Pi(R)^3); // _G. C. Greubel_, Oct 26 2018
%Y A091670 Cf. A091671, A091672, A277232, A293238 (inverse), A068466.
%K A091670 nonn,cons
%O A091670 1,2
%A A091670 _Eric W. Weisstein_, Jan 27 2004