A397030 Decimal expansion of Sum_{k>=1} H(k) / (2*k+1)^4, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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

Offset

0,3

Links

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

Ross C. McPhedran and David H. Bailey, New Results for Euler Sums, arXiv:2311.06294 [math.NT], 2023-2025. See p. 19, eq. (106).

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2nd ed., 2023, section 4.1.14, pp. 273-275.

Cornel Ioan Vălean, More (Almost) Impossible Integrals, Sums, and Series, Springer Cham, 2023. See section 4.19, pp. 419-420, eq.(4.101), section 5.19, pp. 460-461, and section 6.19, pp. 595-601.

Index entries for sequences related to harmonic numbers.

Formula

Equals 31*zeta(5)/8 - 21*zeta(2)*zeta(3)/16 - 15*log(2)*zeta(4)/8.

Example

0.016240657850235703095128372905552235813003886805619...

Mathematica

RealDigits[31*Zeta[5]/8 - 21*Zeta[2]*Zeta[3]/16 - 15*Log[2]*Zeta[4]/8, 10, 120, -1][[1]]

Prog

(PARI) 31*zeta(5)/8 - 21*zeta(2)*zeta(3)/16 - 15*log(2)*zeta(4)/8

Crossrefs

Cf. A001008, A002805.

Cf. A002117, A002162, A013662.

Sum_{k>=1} H(k) / (2*k+1)^m: A397028 (m=2), A397029 (m=3), this constant (m=4), A397031 (m=5), A397032 (m=6).

Sequence in context: A239578 A248273 A176396 * A198502 A244858 A383081

Adjacent sequences: A397027 A397028 A397029 * A397031 A397032 A397033

Keyword

nonn,cons

Author

Amiram Eldar, Jun 14 2026

Status

approved