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A244676
Decimal expansion of sum_(n>=1) (H(n)^3/(n+1)^6) where H(n) is the n-th harmonic number.
4
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
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
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved