A393118 Decimal expansion of Sum_{k>=1} (-1)^(k+1) * (H(k)/k)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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

Offset

0,1

Comments

The closed-form expression for this sum was found by De Doelder (1991).

References

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 243, eq. (4.145).

Links

Table of n, a(n) for n=0..104.

Narendra Bhandari and Yogesh Joshi, A new proof of quadratic series of Au-Yeung and explicit evaluation of its alternating sum, Indian Journal of Pure and Applied Mathematics, Vol. 55 (2024), pp. 1251-1260.

P. J. De Doelder, On some series containing psi(x)-psi(y) and (psi(x)-psi(y))^2 for certain values of x and y, Journal of Computational and Applied Mathematics, Vol. 37, No. 1-3 (1991), pp. 125-141. See p. 133, eq. (13).

Seán Mark Stewart, Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers, Tatra Mountains Mathematical Publications, Vol. 77, No. 1 (2020), pp. 73-98; alternative link. See p. 75, eq. (4).

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 4.52 (iii), pp. 310, 326, 506-508.

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.

Index entries for sequences related to harmonic numbers.

Formula

Equals (41/16)*zeta(4) - (7/4)*log(2)*zeta(3) + (1/2)*log(2)^2*zeta(2) - (1/12)*log(2)^4 - 2*Li_4(1/2), where Li_4(z) is the polylogarithm function of order 4.

Example

0.656311551607752204630659246374290998942618547375653...

Mathematica

RealDigits[(41/16)*Zeta[4] - (7/4)*Log[2]*Zeta[3] + (1/2)*Log[2]^2*Zeta[2] - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2], 10, 120][[1]]

Prog

(PARI) (41/16)*zeta(4) - (7/4)*log(2)*zeta(3) + (1/2)*log(2)^2*zeta(2) - (1/12)*log(2)^4 - 2*polylog(4, 1/2)

Crossrefs

Cf. A001008, A002805.

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

Related constants: A218505, A393115, A393116, A393117, A393119, A393120, A393143.

Sequence in context: A246673 A220190 A231738 * A198829 A217170 A064844

Adjacent sequences: A393115 A393116 A393117 * A393119 A393120 A393121

Keyword

nonn,cons,changed

Author

Amiram Eldar, Feb 02 2026

Status

approved