A244676 Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^6) where H(n) is the n-th harmonic number.

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

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 27.

Formula

Equals -37/7560*Pi^6*zeta(3) + zeta(3)^3 - 11/120*Pi^4*zeta(5) + 1/2*Pi^2*zeta(7) + 197/24*zeta(9).

Example

0.02289126788224074913774364071997743746511359015190275216397993401922...

Mathematica

RealDigits[197/24*Zeta[9] - 33/4*Zeta[4]*Zeta[5] - 37/8*Zeta[3]*Zeta[6] + Zeta[3]^3 + 3*Zeta[2]*Zeta[7], 10, 104] // First // Prepend[#, 0]&

Prog

(PARI)  default(realprecision, 100);  -37/7560*Pi^6*zeta(3) + zeta(3)^3 - 11/120*Pi^4*zeta(5) + 1/2*Pi^2*zeta(7) + 197/24*zeta(9) \\ G. C. Greubel, Aug 31 2018
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); -(37/7560)*Pi(R)^6*Evaluate(L, 3) + Evaluate(L, 3)^3 - (11/120)*Pi(R)^4*Evaluate(L, 5) + Pi(R)^2*Evaluate(L, 7)/2 + (197/24)*Evaluate(L, 9); // G. C. Greubel, Aug 31 2018

Crossrefs

Cf. A001008, A002805, A002117, A013663, A013665, A013667, A244667, A244674, A244675.

Sequence in context: A100384 A000023 A333934 * A010584 A131659 A137726

Adjacent sequences: A244673 A244674 A244675 * A244677 A244678 A244679

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jul 04 2014

Status

approved