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Decimal expansion of r_(5,1), a constant which is the residue at -4 of the distribution function of the distance travelled in a 5-step uniform random walk.
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%I #5 Sep 17 2014 11:26:31

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

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

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

%N Decimal expansion of r_(5,1), a constant which is the residue at -4 of the distribution function of the distance travelled in a 5-step uniform random walk.

%H Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, <a href="http://dx.doi.org/10.4153/CJM-2011-079-2">Densities of Short Uniform Random Walks</a> p. 974, Canad. J. Math. 64(2012), 961-990.

%F r_(5,1) = 13/225*r_(5,0) - 2/(5*Pi^4*r_(5,0)), where r_(5,0) is A244995 (residue at -2).

%F r_(5,1) = 13/(1800*sqrt(5))*Gamma(1/15)*Gamma(2/15)*Gamma(4/15)*Gamma(8/15)/Pi^4 - 1/sqrt(5)*Gamma(7/15)*Gamma(11/15)*Gamma(13/15)*Gamma(14/15)/Pi^4.

%e 0.0066167302594300817140577380007496562495510327524833...

%t r[5, 0] = (2*Sqrt[15]*Re[HypergeometricPFQ[{1/2, 1/2, 1/2}, {5/6, 7/6}, 125/4]])/Pi^2; r[5, 1] = 13/225*r[5, 0] - 2/(5*Pi^4*r[5, 0]); Join[{0, 0}, RealDigits[r[5, 1], 10, 101] // First]

%Y Cf. A244993, A244994, A244995, A244996, A244997, A244999, A245025, A245026.

%K nonn,cons,walk

%O 0,3

%A _Jean-François Alcover_, Sep 17 2014