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.

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, 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, 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

Offset

0,1

References

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 245, eq. (4.149).

K. Ramachandra and R. Sitaramachandrarao, On series, integrals and continued fractions - II, Madras Univ. J., Sect. B, 51 (1988), pp. 181-198.

Links

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

K. Ramachandra, On series integrals and continued fractions I, Hardy-Ramanujan Journal, Vol. 4 (1981), pp. 1-11.

K. Ramachandra, On series, integrals and continued fractions, III, Acta Arithmetica, Vol. 99, No. 3 (2001), pp. 257-266.

Formula

Equals (Pi*log(2))^2/8 + 5*zeta(4)/8 - 9*zeta(3)*log(2)/8 - log(2)^4/4.

Example

0.27412574654925297067883303678750470762654489295575...

Mathematica

RealDigits[(Pi*Log[2])^2/8 + 5*Zeta[4]/8 - 9*Zeta[3]*Log[2]/8 - Log[2]^4/4, 10, 120][[1]]

Prog

(PARI) (Pi*log(2))^2/8 + 5*zeta(4)/8 - 9*zeta(3)*log(2)/8 - log(2)^4/4

Crossrefs

Cf. A001008, A002805.

Cf. A000796, A002117, A002162, A013662.

Related constants: A152648, A152649, A152651, A218505, A233090, A238168, A238181, A238182, A241753, A244667, A253191, A256988, A345203.

Sequence in context: A016592 A371876 A051558 * A353908 A144805 A198745

Adjacent sequences: A384455 A384456 A384457 * A384459 A384460 A384461

Keyword

nonn,cons,easy

Author

Amiram Eldar, May 30 2025

Status

approved