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A384458 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*H(k)^3/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

%I A384458 #5 May 30 2025 10:35:17
%S A384458 2,7,4,1,2,5,7,4,6,5,4,9,2,5,2,9,7,0,6,7,8,8,3,3,0,3,6,7,8,7,5,0,4,7,
%T A384458 0,7,6,2,6,5,4,4,8,9,2,9,5,5,7,5,2,9,6,5,4,7,1,8,1,4,6,2,7,5,5,3,2,1,
%U A384458 6,0,6,7,5,8,7,1,4,1,9,7,0,1,0,3,5,8,3,7,2,2,3,8,6,9,4,8,6,6,3,0,7,0,4,6,6
%N A384458 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*H(k)^3/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
%D A384458 Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 245, eq. (4.149).
%D A384458 K. Ramachandra and R. Sitaramachandrarao, On series, integrals and continued fractions - II, Madras Univ. J., Sect. B, 51 (1988), pp. 181-198.
%H A384458 K. Ramachandra, <a href="https://doi.org/10.46298/hrj.1981.93">On series integrals and continued fractions I</a>, Hardy-Ramanujan Journal, Vol. 4 (1981), pp. 1-11.
%H A384458 K. Ramachandra, <a href="https://doi.org/10.4064/aa99-3-3">On series, integrals and continued fractions, III</a>, Acta Arithmetica, Vol. 99, No. 3 (2001), pp. 257-266.
%F A384458 Equals (Pi*log(2))^2/8 + 5*zeta(4)/8 - 9*zeta(3)*log(2)/8 - log(2)^4/4.
%e A384458 0.27412574654925297067883303678750470762654489295575...
%t A384458 RealDigits[(Pi*Log[2])^2/8 + 5*Zeta[4]/8 - 9*Zeta[3]*Log[2]/8 - Log[2]^4/4, 10, 120][[1]]
%o A384458 (PARI) (Pi*log(2))^2/8 + 5*zeta(4)/8 - 9*zeta(3)*log(2)/8 - log(2)^4/4
%Y A384458 Cf. A001008, A002805.
%Y A384458 Cf. A000796, A002117, A002162, A013662.
%Y A384458 Related constants: A152648, A152649, A152651, A218505, A233090, A238168, A238181, A238182, A241753, A244667, A253191, A256988, A345203.
%K A384458 nonn,cons,easy
%O A384458 0,1
%A A384458 _Amiram Eldar_, May 30 2025