A258982 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,3).

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

Offset

0,2

Links

Table of n, a(n) for n=0..99.

Richard E. Crandall and Joe P. Buhler, On the evaluation of Euler Sums, Exp. Math. 3 (4) (1994) 275-285 Table 1.

Eric Weisstein's World of Mathematics, Multivariate Zeta Function

Wikipedia, Multiple zeta function

Formula

Equals Sum_{m>=2} Sum_{n=1..m-1} 1/(m^5*n^3).

Equals Sum_{m>=2} H(m-1, 3)/m^5, where H(n,3) is the n-th harmonic number of order 3. [simplified by Vaclav Kotesovec, Oct 18 2025]

Equals Sum_{m>=2} (polygamma(2,m)+zeta(3))/(2*m^5).

Equals 5*zeta(3)*zeta(5) - (147/24)*zeta(8) - (5/2)*zetamult(6, 2), where zetamult(6,2) is A258947.

Example

0.03770767298484754401130478229365991482260131941527752401264507780391...

Mathematica

digits = 99; zetamult[6, 2] = NSum[HarmonicNumber[m-1, 2]/m^6, {m, 2, Infinity}, WorkingPrecision -> digits+20, NSumTerms -> 200, Method -> {"NIntegrate", "MaxRecursion" -> 18}]; zetamult[5, 3] = 5*Zeta[3]*Zeta[5] - (147/24)*Zeta[8] - (5/2)*zetamult[6, 2]; Join[{0}, RealDigits[zetamult[5, 3], 10, digits] // First]

Prog

(PARI) zetamult([5, 3]) \\ Charles R Greathouse IV, Jan 21 2016

Crossrefs

Cf. A072691, A197110, A258947.

Sequence in context: A013564 A009467 A388762 * A227336 A373862 A353049

Adjacent sequences: A258979 A258980 A258981 * A258983 A258984 A258985

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 16 2015

Status

approved