A393119 Decimal expansion of Sum_{k>=1} (-1)^(k+1) * (AH(k)/k)^2, where AH(K) = A058313(k)/A058312(k) is the k-th alternating harmonic (or skew-harmonic) number.

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

Offset

1,3

Links

Table of n, a(n) for n=1..105.

Cornel Ioan Vălean, Calculating the Skew-Harmonic Number Versions of the (Alternating) Au-Yeung Series by Exploiting The Master Theorem of Series, ResearchGate, September 2024.

Ce Xu, Yingyue Yang, and Jianwen Zhang, Explicit evaluation of quadratic Euler sums, International Journal of Number Theory, Vol. 13, No. 3 (2017), pp. 655-672; arXiv preprint, arXiv:1609.04923 [math.NT], 2016. See Example 3.6.

Formula

Equals (1/6)*log(2)^4 + 2*log(2)^2*zeta(2) + (7/4)*log(2)*zeta(3) - (61/16)*zeta(4) + 4*Li_4(1/2), where Li_4(z) is the polylogarithm function of order 4.

Example

1.020762613101267304892103939095903005303665665025061...

Mathematica

RealDigits[(1/6)*Log[2]^4 + 2*Log[2]^2*Zeta[2] + (7/4)*Log[2]*Zeta[3] - (61/16)*Zeta[4] + 4*PolyLog[4, 1/2], 10, 120][[1]]

Prog

(PARI) (1/6)*log(2)^4 + 2*log(2)^2*zeta(2) + (7/4)*log(2)*zeta(3) - (61/16)*zeta(4) + 4*polylog(4, 1/2)

Crossrefs

Cf. A058312, A058313.

Cf. A002117, A002162, A013661, A013662, A099218.

Related constants: A218505, A393115, A393116, A393117, A393118, A393120, A393143.

Sequence in context: A351727 A337450 A228819 * A104540 A390321 A178818

Adjacent sequences: A393116 A393117 A393118 * A393120 A393121 A393122

Keyword

nonn,cons

Author

Amiram Eldar, Feb 02 2026

Status

approved