A397375 - OEIS

%I #4 Jun 25 2026 00:45:22

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

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

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

%N Decimal expansion of Sum_{k>=1} (zeta(2) - H(k,2))*H(k)/(2*k+1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and H(k,2) = A007406(k)/A007407(k) is the k-th generalized harmonic number of order 2.

%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 4.29, p. 431, eq. (4.120), section 5.29, p. 463, section 6.29, pp. 646-651.

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

%F Equals log(2)^4/6 - log(2)^2*zeta(2) - zeta(4)/2 + 4*Li_4(1/2).

%e 0.776913616156587749440029749938859188817810502311313...

%t RealDigits[Log[2]^4/6 - Log[2]^2*Zeta[2] - Zeta[4]/2 + 4*PolyLog[4, 1/2], 10, 120][[1]]

%o (PARI) log(2)^4/6 - log(2)^2*zeta(2) - zeta(4)/2 + 4*polylog(4, 1/2)

%Y Cf. A001008, A002805, A007406, A007407.

%Y Cf. A002162, A013661, A013662, A099218.

%Y Cf. A397376, A397377, A397378.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Jun 23 2026