A363539 Decimal expansion of Sum_{k>=1} (H(k)^2 - (log(k) + gamma)^2)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

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

Offset: 1

Author

Amiram Eldar, Jun 09 2023

Keywords

• nonn
• cons

Comments

The formula for this sum was found by Olivier Oloa and proved by Roberto Tauraso in 2014.

Examples

			1.96896908391052854646489145379668054231137794286819...
		

Links

• 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.
• Olivier Oloa, A closed form of the series Sum_{n=1..oo} (H(n)^2 - (gamma + ln(n))^2)/n, Mathematics StackExchange, 2014.

Crossrefs

Cf. A001008, A001620, A002117, A002805, A082633, A086279, A363538, A363540.

Programs

Mathematica

RealDigits[-StieltjesGamma[2] - 2*EulerGamma*StieltjesGamma[1] - 2*EulerGamma^3/3 + 5*Zeta[3]/3, 10, 120][[1]]

Formula

Equals -gamma_2 - 2*gamma*gamma_1 - (2/3)*gamma^3 + (5/3)*zeta(3), where gamma_1 and gamma_2 are the 1st and 2nd Stieltjes constants (A082633, A086279).