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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).

%I A363539 #5 Jun 09 2023 01:30:52
%S A363539 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,
%T A363539 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,
%U A363539 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
%N 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).
%C A363539 The formula for this sum was found by Olivier Oloa and proved by Roberto Tauraso in 2014.
%H A363539 Iaroslav V. Blagouchine, <a href="https://doi.org/10.1016/j.jnt.2015.06.012">Expansions of generalized Euler's constants into the series of polynomials in Pi^(-2) and into the formal enveloping series with rational coefficients only</a>, Journal of Number Theory, Vol. 158 (2016), pp. 365-396.
%H A363539 Olivier Oloa, <a href="https://math.stackexchange.com/questions/866382/a-closed-form-of-the-series-sum-n-1-infty-frach-n2-gamma-ln-n2">A closed form of the series Sum_{n=1..oo} (H(n)^2 - (gamma + ln(n))^2)/n</a>, Mathematics StackExchange, 2014.
%F A363539 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).
%e A363539 1.96896908391052854646489145379668054231137794286819...
%t A363539 RealDigits[-StieltjesGamma[2] - 2*EulerGamma*StieltjesGamma[1] - 2*EulerGamma^3/3 + 5*Zeta[3]/3, 10, 120][[1]]
%Y A363539 Cf. A001008, A001620, A002117, A002805, A082633, A086279, A363538, A363540.
%K A363539 nonn,cons
%O A363539 1,2
%A A363539 _Amiram Eldar_, Jun 09 2023