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.
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, 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, 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
Offset: 0
Examples
0.00464045020357812398918052923623454672526939078320273923...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
- Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) p. 34.
Programs
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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
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Mathematica
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] -
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
Formula
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.