A233033 - OEIS

%I #12 Aug 27 2014 16:39:11

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

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

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

%N Decimal expansion of sum_(n=1..infinity) (-1)^(n-1)*H(n)/n^3 where H(n) is the n-th harmonic number.

%H Philippe Flajolet, 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.

%F Equals 11*Pi^4/360 +1/12*Pi^2*log(2)^2 -log(2)^4/12 -2*Li4(1/2) -7/4*log(2)*zeta(3).

%F Also, equals 1/2*integral_{z=0..1} (log(z)^2*log(1+z)) / (z*(1+z)) dz.

%e 0.859247157928590615539909939475759980712884350860414926760520689766...

%t RealDigits[ 11*Pi^4/360 + 1/12*Pi^2*Log[2]^2 - Log[2]^4/12 - 2*PolyLog[4, 1/2] - 7/4*Log[2]*Zeta[3], 10, 100] // First

%o (PARI) 11*Pi^4/360 + Pi^2*log(2)^2/12 - log(2)^4/12 - 2*polylog(4, 1/2) - 7*log(2)*zeta(3)/4 \\ _Charles R Greathouse IV_, Aug 27 2014

%Y Cf. A076788 (same alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3).

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Dec 03 2013