A396743 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*H(k)*H(k,2)/k^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and H(k,2) = A007406(k)/A007407(k) is the k-th generalized harmonic number of order 2.

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

Offset

0,1

References

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 249, eq. (4.157).

Links

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

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 4.57, p. 312, eq. (4.95), section 5.57, p. 328, section 6.57, pp. 522-523.

Index entries for sequences related to harmonic numbers.

Formula

Equals 23*zeta(5)/8 - 7*log(2)^2*zeta(3)/4 + 2*log(2)^3*zeta(2)/3 + 15*zeta(2)*zeta(3)/16 - 2*log(2)^5/15 - 4*log(2)*polylog(4, 1/2) - 4*polylog(5, 1/2).

Example

0.699719142274147392678652788017124786804957113021163...

Mathematica

RealDigits[23*Zeta[5]/8 - 7*Log[2]^2*Zeta[3]/4 + 2*Log[2]^3*Zeta[2]/3 + 15*Zeta[2]*Zeta[3]/16 - 2*Log[2]^5/15 - 4*Log[2]*PolyLog[4, 1/2] - 4*PolyLog[5, 1/2], 10, 120][[1]]

Prog

(PARI) 23*zeta(5)/8 - 7*log(2)^2*zeta(3)/4 + 2*log(2)^3*zeta(2)/3 + 15*zeta(2)*zeta(3)/16 - 2*log(2)^5/15 - 4*log(2)*polylog(4, 1/2) - 4*polylog(5, 1/2)

Crossrefs

Cf. A001008, A002805, A007406, A007407.

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

Sequence in context: A051496 A330564 A195296 * A273951 A307053 A100403

Adjacent sequences: A396740 A396741 A396742 * A396744 A396745 A396746

Keyword

nonn,cons

Author

Amiram Eldar, Jun 04 2026

Status

approved