A244997 -id:A244997 - cp's OEIS frontend

A244999 Decimal expansion of the moment derivative W_5'(0) associated with the radial probability distribution of a 5-step uniform random walk.

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

Offset: 0

Jean-François Alcover, Jul 09 2014

			0.54441256175218558519587806274502767666605280202852744228702849390214369...

Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks p. 978, Canad. J. Math. 64(2012), 961-990.

Cf. A244996, A244997.

digits = 92; Log[2] - EulerGamma - NIntegrate[(BesselJ[0, x]^5 - 1)/x, {x, 0, 1}, WorkingPrecision -> digits + 20] - NIntegrate[BesselJ[0, x]^5/x, {x, 1, Infinity}, WorkingPrecision -> digits + 20] // RealDigits[#, 10, digits] & // First

W_5'(0) = log(2) - gamma - integral_{x=0..1}((J_0(x)^5-1)/x) - integral_{x>1}(J_0(x)^5/x), where J_0 is the Bessel function of the first kind.

A245026 Decimal expansion of the moment derivative W_4'(2) associated with the radial probability distribution of a 4-step uniform random walk.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 10 2014

			3.48925939156196990636753214786793582537972178245158184401911744937604...

Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks p. 978, Canad. J. Math. 64(2012), 961-990.

Cf. A244997, A245025.

RealDigits[3 + (14*Zeta[3] - 12)/Pi^2 , 10, 104] // First

W_4'(2) = 3 + (14*zeta(3) - 12)/Pi^2.

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.

Original entry on oeis.org

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

Jean-François Alcover, Sep 17 2014

			0.0066167302594300817140577380007496562495510327524833...

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.

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

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]

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.

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A244999 Decimal expansion of the moment derivative W_5'(0) associated with the radial probability distribution of a 5-step uniform random walk.

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A245026 Decimal expansion of the moment derivative W_4'(2) associated with the radial probability distribution of a 4-step uniform random walk.

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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.

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