A397028 Decimal expansion of Sum_{k>=1} H(k) / (2*k+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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

Offset

0,1

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. (104).

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 7*zeta(3)/4 - Pi^2*log(2)/4 = 7*zeta(3)/4 - 3*zeta(2)*log(2)/2.

Example

0.393327464565010863237671994156513466013781249548894...

Mathematica

RealDigits[7*Zeta[3]/4 - 3*Zeta[2]*Log[2]/2, 10, 120][[1]]

Prog

(PARI) 7*zeta(3)/4 - Pi^2*log(2)/4

Crossrefs

Cf. A001008, A002805.

Cf. A002117, A002162, A013661.

Sum_{k>=1} H(k) / (2*k+1)^m: this constant (m=2), A397029 (m=3), A397030 (m=4), A397031 (m=5), A397032 (m=6).

Sequence in context: A375503 A201416 A072560 * A290506 A303111 A299633

Adjacent sequences: A397025 A397026 A397027 * A397029 A397030 A397031

Keyword

nonn,cons

Author

Amiram Eldar, Jun 14 2026

Status

approved