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

0,3

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.

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.

Example

0.00464045020357812398918052923623454672526939078320273923...

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]

Prog

(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
(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

Crossrefs

Cf. A244667, A244674, A244675, A244676.

Sequence in context: A253623 A227436 A023901 * A342278 A173678 A219234

Adjacent sequences: A247666 A247667 A247668 * A247670 A247671 A247672

Keyword

nonn,cons

Author

Jean-François Alcover, Sep 22 2014

Status

approved