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A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.

%I A233090 #26 Jul 09 2025 04:37:42
%S A233090 7,5,1,2,8,5,5,6,4,4,7,4,7,4,6,4,2,8,3,7,4,8,3,6,3,5,0,9,4,4,6,5,6,2,
%T A233090 4,4,2,2,8,1,1,6,4,3,2,7,1,2,8,1,1,8,0,1,1,2,0,1,6,9,7,2,2,0,8,8,6,4,
%U A233090 8,8,7,8,6,1,6,4,4,5,6,8,1,3,6,6,5,3,4,9,2,1,0,0,5,8,3,4,5,3,6,3
%N A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.
%D A233090 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.
%H A233090 R. Barbieri, J. A. Mignaco, and E. Remiddi, <a href="https://dx.doi.org/10.1007/BF02728545">Electron form factors up to fourth order. I.</a>, Il Nuovo Cim. 11A (4) (1972) 824-864, table II (13)
%H A233090 Philippe Flajolet and Bruno Salvy, <a href="http://algo.inria.fr/flajolet/Publications/FlSa98.pdf">Euler Sums and Contour Integral Representations</a>, Experimental Mathematics 7:1 (1998), page 32.
%H A233090 Paul J. Nahin, <a href="https://doi.org/10.1007/978-3-030-43788-6">Inside interesting integrals</a>, Undergrad. Lecture Notes in Physics, Springer (2020), (C5.15)
%H A233090 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2007). See p. 561.
%F A233090 Equals 5*zeta(3)/8.
%F A233090 Equals -Integral_{x=0..1} (log(1+x)*log(1-x)/x)*dx. - _Amiram Eldar_, May 06 2023
%F A233090 Equals Sum_{m>=1} Sum_{n>=1} (-1)^(m-1)/(m*n*(m + n)) (see Finch). - _Stefano Spezia_, Nov 02 2024
%e A233090 0.7512855644747464283748363509446562442281164327128118011201697220886...
%t A233090 RealDigits[ 5*Zeta[3]/8, 10, 100] // First
%Y A233090 Cf. A002117 (zeta(3)), A197070 (3*zeta(3)/4), A233091 (7*zeta(3)/8), A076788 (alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3), A233033 (alternating sum with denominator n^3).
%K A233090 nonn,cons
%O A233090 0,1
%A A233090 _Jean-François Alcover_, Dec 04 2013, after the comment by _Peter Bala_ about A233033