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

Author

Jean-François Alcover, Sep 22 2014

Keywords

• nonn
• cons

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.

Crossrefs

Cf. A244667, A244674, A244675, A244676.

Programs

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

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.