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%I #18 Jul 18 2014 15:53:20
%S 1,6,2,6,5,4,6,6,7,3,9,7,4,2,0,0,8,0,7,7,5,5,6,4,5,6,5,1,7,3,5,9,1,1,
%T 0,1,1,8,7,0,6,4,2,0,8,3,3,7,6,5,9,9,2,3,7,2,6,7,6,3,0,6,9,8,3,1,8,4,
%U 3,5,7,7,2,9,8,2,1,0,7,4,9,2,1,6,7,2,0,0,7,4,6,3,7,5,7,4,9,8,1,0,6,7,9,6,9
%N Decimal expansion of the series limit sum_{k>=1} (-1)^(k+1) sum_{t=1..k} 1/(t^2*(k+1)^2).
%C Equals the alternating sum over (-1)^(k+1)*H_k^(2)/(k+1)^2, where H_k^(2) is the harmonic sum over inverse squares, H_k^(2) = sum_{t=1..k} 1/t^2 = 1, 5/4, 49/36, 205/144, 5269/3600,..., see A007406. The sum over H_k^(2)/(k+1)^2, over the absolute values, is Pi^4/120 = 0.811742425283353...
%H D. H. Bailey, J. M. Borwein, R. Girgensohn, <a href="http://www.emis.de/journals/EM/expmath/volumes/3/3.html">Experimental evaluation of Euler sums</a>, Exp. Math. 3 (1994) 17, variable alpha(2,2)
%H G. Rutledge, R. D. Douglass, <a href="http://www.jstor.org/stable/2303745">Table of definite integrals</a>, Am. Math. Monthly 45 (1938) 525, variable A_4.
%F Equals -4*A099218 +13*Pi^4/288 -7*A002117*log(2)/2+log^2(2)*(Pi^2-log^2(2))/6.
%e 0.162654667397420080...
%p a099218 := polylog(4,1/2) ;
%p -4*a099218+13*Pi^4/288-7/2*Zeta(3)*log(2)+Pi^2/6*(log(2))^2-(log(2))^4/6 ;
%p evalf(%) ;
%t NSum[(-1)^(k + 1)*HarmonicNumber[k, 2]/(k + 1)^2, {k, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 110] // RealDigits[#, 10, 105] & // First (* or, from formula: *) 13*Pi^4/288 + 1/6*Pi^2*Log[2]^2 - 1/6*Log[2]*(Log[2]^3 + 21*Zeta[3]) - 4*PolyLog[4, 1/2] // RealDigits[#, 10, 105]& // First (* _Jean-François Alcover_, Mar 06 2013 *)
%o (PARI) 13*Pi^4/288 + 1/6*Pi^2*log(2)^2 - 1/6*log(2)*(log(2)^3 + 21*zeta(3)) - 4*polylog(4, 1/2) \\ _Charles R Greathouse IV_, Jul 18 2014
%K cons,nonn,easy
%O 0,2
%A _R. J. Mathar_, Jul 19 2012
%E More terms from _Jean-François Alcover_, Feb 12 2013
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