A396735 Decimal expansion of Sum_{k>=1} H(k)/((k+1)^4*2^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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

Offset

0,2

Links

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

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 4.51, p. 309, eq. (4.83), section 5.51, p. 326, section 6.51, pp. 498-502.

Index entries for sequences related to harmonic numbers.

Formula

Equals zeta(5)/16 - log(2)*zeta(4)/4 + log(2)^2*zeta(3) - log(2)^3*zeta(2)/3 - zeta(2)*zeta(3) + log(2)^5/20 + 2*log(2)*Li_4(1/2) + 2*Li_5(1/2).

Example

0.037061572130933226084716510898886039678706583286763...

Mathematica

RealDigits[Zeta[5]/16 - Log[2]*Zeta[4]/4 + Log[2]^2*Zeta[3] - Log[2]^3*Zeta[2]/3 - Zeta[2]*Zeta[3] + Log[2]^5/20 + 2*Log[2]*PolyLog[4, 1/2] + 2*PolyLog[5, 1/2], 10, 120, -1][[1]]

Prog

(PARI) zeta(5)/16 - log(2)*zeta(4)/4 + log(2)^2*zeta(3) - log(2)^3*zeta(2)/3 - zeta(2)*zeta(3) + log(2)^5/20 + 2*log(2)*polylog(4, 1/2) + 2*polylog(5, 1/2)

Crossrefs

Cf. A001008, A002805.

Cf. A002117, A002162, A013662, A013663, A099218, A099219.

Sum_{k>=1} H(k)/((k+1)^m*2^k): A016627 (m=0), A253191 (m=1), A395618 (m=2), A396733 (m=3), this constant (m=4).

Sequence in context: A291835 A197835 A197005 * A199778 A369381 A086729

Adjacent sequences: A396727 A396733 A396734 * A396736 A396737 A396738

Keyword

nonn,cons

Author

Amiram Eldar, Jun 04 2026

Status

approved