A398080 Decimal expansion of Sum_{k>=1} (zeta(3) - H(2*k,3)) * H(k)/(2*k+1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and H(k,3) = A007408(k)/A007409(k) is the k-th harmonic number of order 3.

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

Offset

0,2

Links

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

Cornel Ioan Vălean, More (Almost) Impossible Integrals, Sums, and Series, Springer Cham, 2023. See section 4.40, p. 437, eq. (4.135), section 5.40, p. 465, section 6.40, pp. 690-694.

Index entries for sequences related to harmonic numbers.

Formula

Equals log(2)^5/15 - log(2)^3*zeta(2)/3 - 7*log(2)^2*zeta(3)/8 + 15*log(2)*zeta(4)/32 - 31*zeta(5)/64 - 7*zeta(2)*zeta(3)/16 + 2*log(2)*Li_4(1/2) + 2*Li_5(1/2).

Example

0.041232806279660023207810779913235512461326027149676...

Mathematica

RealDigits[Log[2]^5/15 - Log[2]^3*Zeta[2]/3 - 7*Log[2]^2*Zeta[3]/8 + 15*Log[2]*Zeta[4]/32 - 31*Zeta[5]/64 - 7*Zeta[2]*Zeta[3]/16 + 2*Log[2]*PolyLog[4, 1/2] + 2*PolyLog[5, 1/2], 10, 120, -1][[1]]

Prog

(PARI) log(2)^5/15 - log(2)^3*zeta(2)/3 - 7*log(2)^2*zeta(3)/8 + 15*log(2)*zeta(4)/32 - 31*zeta(5)/64 - 7*zeta(2)*zeta(3)/16 + 2*log(2)*polylog(4, 1/2) + 2*polylog(5, 1/2)

Crossrefs

Cf. A001008, A002805, A007408, A007409.

Cf. A002117, A002162, A013661, A013662, A013663, A099218, A099219.

Sequence in context: A375576 A064124 A106654 * A252094 A196138 A198185

Adjacent sequences: A398077 A398078 A398079 * A398081 A398082 A398083

Keyword

nonn,cons,new

Author

Amiram Eldar, Jul 19 2026

Status

approved