A396741 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*H(k)^2/k^3, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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

Offset

0,1

References

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 247, eq. (4.153).

Links

Table of n, a(n) for n=0..105.

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 4.55, p. 311, eq. (4.93), section 5.55, p. 327, section 6.55, pp. 517-520.

Index entries for sequences related to harmonic numbers.

Formula

Equals 2*log(2)^5/15 - 11*zeta(2)*zeta(3)/8 - 19*zeta(5)/32 + 7*log(2)^2*zeta(3)/4 - 2*log(2)^3*zeta(2)/3 + 4*log(2)*polylog(4, 1/2) + 4*polylog(5, 1/2).

Example

0.800701645891598200053549847313241971289317467928766...

Mathematica

RealDigits[2*Log[2]^5/15 - 11*Zeta[2]*Zeta[3]/8 - 19*Zeta[5]/32 + 7*Log[2]^2*Zeta[3]/4 - 2*Log[2]^3*Zeta[2]/3 + 4*Log[2]*PolyLog[4, 1/2] + 4*PolyLog[5, 1/2], 10, 120][[1]]

Prog

(PARI) 2*log(2)^5/15 - 11*zeta(2)*zeta(3)/8 - 19*zeta(5)/32 + 7*log(2)^2*zeta(3)/4 - 2*log(2)^3*zeta(2)/3 + 4*log(2)*polylog(4, 1/2) + 4*polylog(5, 1/2)

Crossrefs

Cf. A001008, A002805.

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

Sequence in context: A246560 A199064 A280653 * A274413 A369885 A167261

Adjacent sequences: A396738 A396739 A396740 * A396742 A396743 A396744

Keyword

nonn,cons

Author

Amiram Eldar, Jun 04 2026

Status

approved