A396742 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*H(k)^3/k^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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

Offset

0,1

References

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 251, eq. (4.160).

Links

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

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 4.56, p. 312, eq. (4.94), section 5.56, p. 327, section 6.56, pp. 520-521.

Index entries for sequences related to harmonic numbers.

Formula

Equals log(2)^5/5 - log(2)^3*zeta(2) + 21*log(2)^2*zeta(3)/8 - 27*zeta(2)*zeta(3)/16 - 9*zeta(5)/4 + 6*log(2)*polylog(4, 1/2) + 6*polylog(5, 1/2).

Example

0.532967933050864850797531853374254224262188626942787...

Mathematica

RealDigits[Log[2]^5/5 - Log[2]^3*Zeta[2] + 21*Log[2]^2*Zeta[3]/8 - 27*Zeta[2]*Zeta[3]/16 - 9*Zeta[5]/4 + 6*Log[2]*PolyLog[4, 1/2] + 6*PolyLog[5, 1/2], 10, 120][[1]]

Prog

(PARI) log(2)^5/5 - log(2)^3*zeta(2) + 21*log(2)^2*zeta(3)/8 - 27*zeta(2)*zeta(3)/16 - 9*zeta(5)/4 + 6*log(2)*polylog(4, 1/2) + 6*polylog(5, 1/2)

Crossrefs

Cf. A001008, A002805.

Cf. A002117, A002162, A013661, A013663, A099218, A099219.

Sequence in context: A171025 A222362 A352024 * A176524 A309276 A268690

Adjacent sequences: A396739 A396740 A396741 * A396743 A396744 A396745

Keyword

nonn,cons

Author

Amiram Eldar, Jun 04 2026

Status

approved