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%I #22 May 06 2020 01:48:48
%S 1,5,6,1,6,6,9,3,3,3,8,1,1,7,6,9,1,5,8,8,1,0,3,5,9,0,9,6,8,7,9,8,8,1,
%T 9,3,6,8,5,7,7,6,7,0,9,8,4,0,3,0,3,8,7,2,9,5,7,5,2,9,3,5,4,4,9,7,0,7,
%U 5,0,3,7,4,4,0,2,9,5,7,9,1,4,5,5,2,0,5,6,5,3,7,0,9,3,5,8,1,4,7,5
%N Decimal expansion of Euler's Multi-Zeta Sum S(2,3) = Sum_{n >= 1} (Sum_{k=1..n}((-1)^(k + 1)/k)^2/(n + 1)^3).
%H D. H. Bailey and J. M. Borwein, <a href="http://escholarship.org/uc/item/6b6986dn#page-10">Euler's Multi-Zeta Sums</a>, Lawrence Berkeley National Laboratory, 1995, p. 10.
%e 0.1561669333811769158810359096879881936857767...
%t 4*PolyLog[5, 1/2] - 1/30*Log[2]^5 - 17/32*Zeta[5] - 11/720*Pi^4*Log[2] + 7/4*Zeta[3]*Log[2]^2 + 1/18*Pi^2*Log[2]^3 - 1/8*Pi^2*Zeta[3] // RealDigits[#, 10, 100]& // First
%o (PARI) 4*polylog(5,1/2)-1/30*log(2)^5-17/32*zeta(5) - 11/720*Pi^4*log(2) + 7/4*zeta(3)*log(2)^2 + 1/18*Pi^2*log(2)^3 - 1/8*Pi^2*zeta(3) \\ _Charles R Greathouse IV_, Jul 18 2014
%o (SageMath)
%o RR = RealBallField(380)
%o f = fast_callable(4*polylog(5, 1/2) - 1/30*log(2)^5 - 17/32*zeta(5) - 11/720*pi^4*log(2) + 7/4*zeta(3)*log(2)^2 + 1/18*pi^2*log(2)^3 - 1/8*pi^2*zeta(3), vars=[x], domain=RR)
%o print([int(t) for t in str(f(0))[3:103]]) # _Peter Luschny_, May 06 2020
%Y Cf. A218505.
%K nonn,cons
%O 0,2
%A _Jean-François Alcover_, Feb 18 2014
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