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Decimal expansion of Sum_{k,m>=1} (-1)^(k+m) * H(k) * H(m) / (k+m+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
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%I #9 Oct 24 2024 10:44:52

%S 3,2,3,0,7,2,9,9,7,2,9,6,1,0,1,9,5,5,8,5,5,0,1,5,8,9,7,5,6,3,7,3,9,3,

%T 5,6,9,0,0,6,5,5,7,4,4,7,2,6,6,8,4,8,7,7,2,1,6,6,8,6,4,8,7,4,6,2,6,9,

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

%N Decimal expansion of Sum_{k,m>=1} (-1)^(k+m) * H(k) * H(m) / (k+m+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

%H Paul Bracken, <a href="https://cms.math.ca/publications/crux/issue/?volume=50&amp;issue=3">Problem 4927</a>, Crux Mathematicorum, Vol. 50, No. 3 (March, 2024), p. 149; Theo Koupelis, <a href="https://cms.math.ca/publications/crux/issue/?volume=50&amp;issue=8">Solution to Problem 4927</a>, ibid., Vol. 50, No. 8 (Oct. 2024), pp. 423-426.

%F Equals log(2)^3/3 + log(2)^2 + 2*log(2) - zeta(2) - zeta(3)/4.

%e 0.032307299729610195585501589756373935690065574472668...

%t RealDigits[Log[2]^3/3 + Log[2]^2 + 2*Log[2] - Zeta[2] - Zeta[3]/4, 10, 120][[1]]

%o (PARI) log(2)^3/3 + log(2)^2 + 2*log(2) - zeta(2) - zeta(3)/4

%Y Cf. A001008, A002805.

%Y Cf. A002117, A002162, A013661, A233090, A253191, A255986.

%K nonn,cons,easy

%O -1,1

%A _Amiram Eldar_, Oct 24 2024