A099218 -id:A099218 - cp's OEIS frontend

A244839 Decimal expansion of the Euler double sum sum_(m>0)(sum_(n>0)((-1)^(m+n-1)/((2m-1)(m+n-1)^3))).

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

Offset: 0

Jean-François Alcover, Jul 07 2014

The computation of this constant is given by Bailey & Borwein as an example of the use of CAS packages to check digital integrity of published mathematics.

			0.87292928952035451895794199102873253738299452053432445689371621121704773167...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. H. Bailey and J. M. Borwein, Experimental computation as an ontological game changer, 2014, see p. 4.
J. M. Borwein, I.J. Zucker and J. Boersma, The evaluation of character Euler double sums, The Ramanujan Journal, April 2008, Volume 15, Issue 3, pp 377-405, see p. 17.
Eric Weisstein's MathWorld, Polylogarithm.

Cf. A099218.

4*PolyLog[4, 1/2] - 151/2880*Pi^4 - Pi^2/6*Log[2]^2 + 1/6*Log[2]^4 + 7/2*Log[2]*Zeta[3] // RealDigits[#, 10, 105]& // First

151*Pi^4/2880 + Pi^2*log(2)^2/6 - 4*polylog(4, 1/2) - log(2)^4/6 - 7*log(2)*zeta(3)/2 \\ Charles R Greathouse IV, Aug 27 2014

4*polylog(4, 1/2) - 151/2880*Pi^4 - Pi^2/6*log(2)^2 + 1/6*log(2)^4 + 7/2*log(2)*zeta(3).

A377647 Decimal expansion of Pi^4/15 + Pi^2log(2)^2/4 - log(2)^4/4 - 21log(2)zeta(3)/4 - 6Li_4(1/2).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 03 2024

			0.142514197935710915708721501209652160895463410764...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.8, p. 47.

Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 181-182.

Cf. A002117, A002162, A002388, A092425, A099218.

RealDigits[Pi^4/15+Pi^2Log[2]^2/4-Log[2]^4/4-21Log[2]Zeta[3]/4-6PolyLog[4,1/2],10,100][[1]]

Integral_{x=1..2} log(x)^3/(x - 1) [Levin] (see Finch).

Integral_{x=0..1} log(1 + x)^3/x (see Shamos).

cp's OEIS Frontend

A244839 Decimal expansion of the Euler double sum sum_(m>0)(sum_(n>0)((-1)^(m+n-1)/((2m-1)(m+n-1)^3))).

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A377647 Decimal expansion of Pi^4/15 + Pi^2log(2)^2/4 - log(2)^4/4 - 21log(2)zeta(3)/4 - 6Li_4(1/2).

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cp's OEIS Frontend

A244839 Decimal expansion of the Euler double sum sum_(m>0)(sum_(n>0)((-1)^(m+n-1)/((2m-1)(m+n-1)^3))).

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Formula

A377647 Decimal expansion of Pi^4/15 + Pi^2*log(2)^2/4 - log(2)^4/4 - 21*log(2)*zeta(3)/4 - 6*Li_4(1/2).

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A377647 Decimal expansion of Pi^4/15 + Pi^2log(2)^2/4 - log(2)^4/4 - 21log(2)zeta(3)/4 - 6Li_4(1/2).