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A240264 Decimal expansion of Sum_{n >= 1} (-1)^(n+1)*H(n,2)/n^2, where H(n,2) is the n-th harmonic number of order 2.

%I A240264 #12 Feb 16 2025 08:33:21
%S A240264 6,3,1,9,6,6,1,9,7,8,3,8,1,6,7,9,0,6,6,6,2,4,4,8,2,3,2,0,1,5,2,7,5,3,
%T A240264 1,8,1,5,6,6,7,1,3,7,1,6,5,8,1,7,2,7,5,5,5,1,5,2,6,0,5,6,7,9,6,5,4,1,
%U A240264 1,7,6,9,2,0,9,4,1,5,6,9,6,2,9,4,2,9,3,3,6,4,7,8,5,5,6,9,1,4,3,0
%N A240264 Decimal expansion of Sum_{n >= 1} (-1)^(n+1)*H(n,2)/n^2, where H(n,2) is the n-th harmonic number of order 2.
%D A240264 B. C. Berndt, Ramanujan's Notebooks Part I, Springer-Verlag,
%H A240264 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%F A240264 Equals zeta(3) - Pi^2/12*log(2).
%F A240264 Let a(p,q) = Sum_{n >= 1} (-1)^(n+1)*H(n,p)/n^q, then A076788 is a(1,1), A233090 is a(1,2) and this sequence is a(2,1).
%F A240264 Equals Sum_{n >= 1} (1/2)^n * H(n,1)/n^2, where H(n,1) = Sum_{k = 1..n} 1/k. See Berndt, p. 258. - _Peter Bala_, Oct 28 2021
%e A240264 0.631966197838...
%t A240264 Zeta[3] - Pi^2/12*Log[2] // RealDigits[#, 10, 100]& // First
%o A240264 (PARI) zeta(3)-log(2)*Pi^2/12 \\ _Charles R Greathouse IV_, Apr 03 2014
%Y A240264 Cf. A076788, A233090.
%K A240264 nonn,cons
%O A240264 0,1
%A A240264 _Jean-François Alcover_, Apr 03 2014