A379827 Decimal expansion of the alternating double sum zeta(-5,-1) = Sum_{i>=2} (Sum_{j=1..i-1} (-1)^(i+j)/(i^5*j)) (negated).

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

Offset

-1,1

Comments

Also known in literature as U(5,1) see e.g. Broadhurst p.13.

Links

Table of n, a(n) for n=-1..103.

D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.

Kam Cheong Au, Evaluation of One-dimensional Polylogarithmic Integral, with Applications to Infinite Series, arXiv:2007.03957 [math.NT], 2020 p.16.

Ming Hao Zhao, MZV evaluation of hypergeometric series, arXiv:2007.02508 [math.NT], 2020.

Artur Jasinski, Complete list of formulas of the multivariate zeta of weight 6 and length 2 [18 of convergent cases]

Formula

Equals -19*Pi^6/42120 - Pi^4*log(2)^2/156 + Pi^2*log(2)^4/208 - log(2)^6/208 + Li(6,-1/8)/39 - 81*Li(6,1/4)/208 + 36*Li(6,1/2)/13 - 3*zeta(3)^2/8 where Li are polylogarithms.

Equals -C/8 - 2*Pi^6/315 - 7*Pi^4*log(2)^2/720 + Pi^2*log(2)^4/72 - 19*log(2)^6/1440 + 3*log(2)*Li(5,1/2) + 8*Li(6,1/2) - log(2)^3*zeta(3)/6 - zeta(3)^2/4 + 45*log(2)*zeta(5)/16 where constant C is HypergeometricPFQ[{1, 1, 1, 1, 1, 1, 1}, {3/2, 2, 2, 2, 2, 2}, -(1/8)] = 0.99742928532464545279962...

Equals zeta(-3,3)/6 + 761*Pi^6/362880 - 27*zeta(3)^2/64 - 31*log(2)*zeta(5)/16.

Equals -zeta(3,-3)/6 + 823*Pi^6/362880 - 35*zeta(3)^2/64 - 31*log(2)*zeta(5)/16.

Equals zeta(4,-2)/4 + 1651*Pi^6/725760 - 33*zeta(3)^2/64 - 31*log(2)*zeta(5)/16.

Equals -zeta(-4,2)/4 + 1697*Pi^6/725760 - 9*zeta(3)^2/16 - 31*log(2)*zeta(5)/16.

Equals zeta(2,-4)/4 + 361*Pi^6/145152 - 9*zeta(3)^2/16 - 31*log(2)*zeta(5)/16.

Equals -zeta(-2,4)/4 + 1669*Pi^6/725760 - 33*zeta(3)^2/64 - 31*log(2)*zeta(5)/16.

Because all above zeta do not contain terms with zata(5) that mean that this constant to contain term - 31*log(2)*zeta(5)/16.

Equals zeta(-5,1) + 13*Pi^6/5040 - 3*zeta(3)^2/4 - 31*log(2)*zeta(5)/16.

Equals zeta(-1,-5) + Pi^6/945 - 15*log(2)*zeta(5)/16.

Equals -1993*Pi^6/1451520 + zeta(3,-3,-3)/(6*zeta(3)) - zeta(5,-1,-3)/zeta(3) - zeta(3,-1,-5)/zeta(3) + 13*zeta(3)^2/128 + 105*log(2)*zeta(5)/64 - 553*Pi^4*zeta(5)/(4608*zeta(3)) - 4445*Pi^2*zeta(7)/(512*zeta(3)) + 226369*zeta(9)/(2304*zeta(3)).

Example

-0.0299016355811494259428679217491..

Mathematica

RealDigits[-19 Pi^6/42120 - Pi^4 Log[2]^2/156 + Pi^2 Log[2]^4/208 - Log[2]^6/208 + PolyLog[6, 1/64]/1248 - PolyLog[6, 1/8]/39 - 81 PolyLog[6, 1/4]/208 + 36/13 PolyLog[6, 1/2] - 3 Zeta[3]^2/8, 10, 105]
(*or*)
RealDigits[-((2 Pi^6)/315) - 1/8 HypergeometricPFQ[{1, 1, 1, 1, 1, 1, 1}, {3/2, 2, 2, 2, 2, 2}, -(1/8)] - 7/720 Pi^4 Log[2]^2 + 1/72 Pi^2 Log[2]^4 - (19 Log[2]^6)/1440 + 3 Log[2] PolyLog[5, 1/2] + 8 PolyLog[6, 1/2] - 1/6 Log[2]^3 Zeta[3] - Zeta[3]^2/4 + 45/16 Log[2] Zeta[5], 10, 105]

Prog

(PARI) polylogmult([5, 1], [-1, -1])

Crossrefs

Cf. A076788 (-zeta(-1,-1)), A197110 (zeta(2,2)), A214508 (zeta(-2,2)), A255685 (-zeta(-3,-1)), A379826 (-zeta(-2,-1)), A379829 (-zeta(-5,-3)).

Sequence in context: A073927 A198358 A145427 * A348637 A104954 A389930

Adjacent sequences: A379824 A379825 A379826 * A379828 A379829 A379830

Keyword

cons,nonn

Author

Artur Jasinski, Jan 03 2025

Status

approved