A398080 - OEIS

%I #5 Jul 19 2026 09:37:12

%S 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,

%T 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,

%U 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

%N 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.

%H Cornel Ioan Vălean, <a href="https://doi.org/10.1007/978-3-031-21262-8">More (Almost) Impossible Integrals, Sums, and Series</a>, Springer Cham, 2023. See section 4.40, p. 437, eq. (4.135), section 5.40, p. 465, section 6.40, pp. 690-694.

%H <a href="/index/Ha#harmonic">Index entries for sequences related to harmonic numbers</a>.

%F 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).

%e 0.041232806279660023207810779913235512461326027149676...

%t 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]]

%o (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)

%Y Cf. A001008, A002805, A007408, A007409.

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

%K nonn,cons,new

%O 0,2

%A _Amiram Eldar_, Jul 19 2026