A247447 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.
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, 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, 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
Offset: 0
Examples
0.0066167302594300817140577380007496562495510327524833...
Links
- Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks p. 974, Canad. J. Math. 64(2012), 961-990.
Programs
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Mathematica
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]
Formula
r_(5,1) = 13/225*r_(5,0) - 2/(5*Pi^4*r_(5,0)), where r_(5,0) is A244995 (residue at -2).
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.