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%I #7 Jul 09 2015 05:47:23
%S 1,6,9,5,5,7,1,7,6,9,9,7,4,0,8,1,8,9,9,5,2,4,1,9,6,5,4,9,6,5,1,5,3,4,
%T 2,1,3,1,6,9,6,9,5,8,1,6,7,2,1,4,2,2,6,0,3,0,7,0,6,8,1,1,0,6,6,7,3,8,
%U 8,6,9,7,1,5,0,3,2,6,3,1,6,3,1,3,7,9,5,6,6,2,9,8,9,7,5,5,8,6,1,7,5,5,0
%N Decimal expansion of the infinite double sum S = Sum_{m>=1} (Sum_{n>=1} 1/(m^2*n*(m+n)^3)).
%H StackExchange, <a href="http://math.stackexchange.com/questions/424807">Integral of polylogarithms and logs in closed form</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Polygamma_function">Polygamma Function</a>.
%F S = (7/4)*zeta(6) - zeta(3)^2/2 - sum_{m>=1} (PolyGamma(1, m+1)/m^4) + (1/2)*sum_{m>=1} (PolyGamma(2, m+1)/m^3), where sum_{m>=1} (PolyGamma(1, m+1)/m^4) is A258989, the second sum being A259927.
%F S simplifies to zeta(6)/6 = Pi^6/5670.
%F 2*A258987 + 6*S = zeta(3)^2.
%e 0.16955717699740818995241965496515342131696958167214226030706811...
%t RealDigits[Pi^6/5670, 10, 103] // First
%o (PARI) Pi^6/5670 \\ _Michel Marcus_, Jul 09 2015
%Y Cf. A258987, A258989, A259927.
%K nonn,cons,easy
%O 0,2
%A _Jean-François Alcover_, Jul 09 2015
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