A256987 Decimal expansion of Sum_{k>=1} H(k)*H(k,2)/k^2 where H(k) is the k-th harmonic number and H(k,2) the k-th harmonic number of order 2.

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

Offset

1,1

References

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 222, eq. (4.107).

Links

Table of n, a(n) for n=1..105.

Donal F. Connon, Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume I, arXiv:0710.4022 [math.HO], 2007-2008. See p. 153.

Alois Panholzer and Helmut Prodinger, Computer-free evaluation of an infinite double sum via Euler sums, Séminaire Lotharingien de Combinatoire 55 (2005), Article B55a.

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 4.25, p. 293, eq. (4.33), section 5.25, p. 320, section 6.25, pp. 398-401.

Eric Weisstein's MathWorld, Harmonic Number.

Index entries for sequences related to harmonic numbers.

Formula

Equals zeta(5) + zeta(2)*zeta(3) = zeta(5) + (Pi^2/6)*zeta(3).

Equals Sum_{k>=1} H(k,2)/k^3 + Sum_{k>=1} H(k,3)/k^2, where H(k,2) = A007406(k)/A007407(k) is the k-th generalized harmonic number of order 2, and H(k,3) = A007408(k)/A007409(k) is the k-th harmonic number of order 3 (Connon, 2007; Olaikhan, 2021, p. 220). - Amiram Eldar, Jun 04 2026

Example

3.01423210544066604452845092794215974013923238616204702067...

Mathematica

RealDigits[Zeta[5] + (Pi^2/6)*Zeta[3], 10, 105] // First

Prog

(PARI) zeta(5) + zeta(2)*zeta(3) \\ Michel Marcus, Apr 14 2015

Crossrefs

Cf. A007406, A007407, A007408, A007409.

Cf. A002117, A013661, A013663, A152648, A152651, A238181, A244667, A256988.

Sequence in context: A226912 A177330 A197126 * A048963 A119458 A106356

Adjacent sequences: A256984 A256985 A256986 * A256988 A256989 A256990

Keyword

nonn,cons,easy,changed

Author

Jean-François Alcover, Apr 14 2015

Status

approved