A244676 - OEIS

%I #14 Sep 08 2022 08:46:08

%S 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,

%T 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,

%U 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

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

%H Vincenzo Librandi, <a href="/A244676/b244676.txt">Table of n, a(n) for n = 0..1000</a>

%H 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) page 27.

%F 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).

%e 0.02289126788224074913774364071997743746511359015190275216397993401922...

%t 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]&

%o (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

%o (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

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

%K nonn,cons,easy

%O 0,2

%A _Jean-François Alcover_, Jul 04 2014