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A247669 Decimal expansion of A_3 = Sum_{n >= 1} H(n)^2/((2n-1)*(2n)*(2n+1))^3, where H(n) is the n-th harmonic number.

%I A247669 #14 Sep 08 2022 08:46:09
%S A247669 0,0,4,6,4,0,4,5,0,2,0,3,5,7,8,1,2,3,9,8,9,1,8,0,5,2,9,2,3,6,2,3,4,5,
%T A247669 4,6,7,2,5,2,6,9,3,9,0,7,8,3,2,0,2,7,3,9,2,3,0,0,4,9,2,4,9,0,7,2,3,4,
%U A247669 4,9,7,1,8,3,2,4,1,2,5,7,3,5,3,7,4,0,5,9,7,0,3,7,2,2,8,3,4,3,8,7,8,1,0,5
%N A247669 Decimal expansion of A_3 = Sum_{n >= 1} H(n)^2/((2n-1)*(2n)*(2n+1))^3, where H(n) is the n-th harmonic number.
%H A247669 G. C. Greubel, <a href="/A247669/b247669.txt">Table of n, a(n) for n = 0..10000</a>
%H A247669 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) p. 34.
%F A247669 A_3 = 4*log(2)^3 + ((7*zeta(3))/8 - 35/4)*log(2)^2 - ((-(1/8))*9*zeta(2) + (7*zeta(3))/8 + (45*zeta(4))/32 - 12)*log(2) - zeta(2)/4 - (41*zeta(3))/8 - (3/32)*zeta(2)*zeta(3) + (45*zeta(4))/64 + (17*zeta(5))/32.
%e A247669 0.00464045020357812398918052923623454672526939078320273923...
%t A247669 A3 = 4*Log[2]^3 + ((7*Zeta[3])/8 - 35/4)*Log[2]^2 - ((-(1/8))*9*Zeta[2] + (7*Zeta[3])/8 + (45*Zeta[4])/32 - 12)*Log[2] - Zeta[2]/4 - (41*Zeta[3])/8 - (3/32)*Zeta[2]*Zeta[3] + (45*Zeta[4])/64 + (17*Zeta[5])/32; Join[{0, 0}, RealDigits[A3, 10, 102] // First]
%o A247669 (PARI) 4*log(2)^3 + ((7*zeta(3))/8 - 35/4)*log(2)^2 - ((-(1/8))*9*zeta(2) + (7*zeta(3))/8 + (45*zeta(4))/32 - 12)*log(2) - zeta(2)/4 - (41*zeta(3))/8 - (3/32)*zeta(2)*zeta(3) + (45*zeta(4))/64 + (17*zeta(5))/32 \\ _Michel Marcus_, Sep 22 2014
%o A247669 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); 4*Log(2)^3 + ((7*Evaluate(L,3))/8 - 35/4)*Log(2)^2 - ((-(1/8))*9*Evaluate(L,2) + (7*Evaluate(L,3))/8 + (45*Evaluate(L,4))/32 - 12)*Log(2) - Evaluate(L,2)/4 - (41*Evaluate(L,3))/8 - (3/32)*Evaluate(L,2)*Evaluate(L,3) + (45*Evaluate(L,4))/64 + (17*Evaluate(L,5))/32; // _G. C. Greubel_, Aug 31 2018
%Y A247669 Cf. A244667, A244674, A244675, A244676.
%K A247669 nonn,cons
%O A247669 0,3
%A A247669 _Jean-François Alcover_, Sep 22 2014