A363540 Decimal expansion of Sum_{k>=1} (H(k)^3 - (log(k) + gamma)^3)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

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

Offset

1,1

Links

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

Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in Pi^(-2) and into the formal enveloping series with rational coefficients only, Journal of Number Theory, Vol. 158 (2016), pp. 365-396.

Formula

Equals -gamma_3 - 3*gamma*gamma_2 - 3*gamma^2*gamma_1 - (3/4)*gamma^4 + (43/8)*zeta(4), where gamma_1, gamma_2 and gamma_3 are the 1st, 2nd and 3rd Stieltjes constants (A082633, A086279, A086280).

Example

5.82174008504864652889686861550204134315033324319577...

Mathematica

RealDigits[-StieltjesGamma[3] - 3*EulerGamma*StieltjesGamma[2] - 3*EulerGamma^2*StieltjesGamma[1] - 3*EulerGamma^4/4 + 43*Zeta[4]/8, 10, 120][[1]]

Crossrefs

Cf. A001008, A001620, A002117, A013662, A082633, A086279, A086280, A363538, A363539.

Sequence in context: A260781 A306068 A021949 * A076788 A195358 A021636

Adjacent sequences: A363537 A363538 A363539 * A363541 A363542 A363543

Keyword

nonn,cons

Author

Amiram Eldar, Jun 09 2023

Status

approved