A258990 -id:A258990 - cp's OEIS frontend

A258987 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,3).

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.213798868224592547099583574508033649640958957865517556144512748947...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function
Index to constants which are multiple zeta values (3,3)

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

RealDigits[(1/2)*Zeta[3]^2 - (1/2)*Zeta[6], 10, 103] // First (* Corrected by Detlef Meya, Jun 06 2025 *)

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

zetamult(3,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^3*n^3)) = (1/2)*zeta(3)^2 - (1/2)*zeta(6). - [Corrected by Detlef Meya, Jun 06 2025 ]

A258983 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

Also zetamult(2, 2, 1). - Charles R Greathouse IV, Jan 04 2017

			0.2288103976033537597687461489416887919325093427198821602294071...

Dominique Manchon, Arborified multiple zeta values, arXiv:1603.01498 [math.CO], 2016.
Jonathan Borwein and Roland Girgensohn, Evaluation of triple Euler Sums, Elec. Jour. of Comb., Vol. 3, Issue 1, 1996. Article R23 (see page 21).
Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Cf. A013663 (zeta(5)), A183699 (zeta(2)*zeta(3)).

RealDigits[3*Zeta[2]*Zeta[3] - (11/2)*Zeta[5], 10, 104] // First

zetamult([3,2]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult([2,2,1]) \\ Charles R Greathouse IV, Jan 04 2017

Equals Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^3*n^2)) = 3*zeta(2)*zeta(3) - (11/2)*zeta(5).

A258989 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,4).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.67452391403396814049156060825742993927838436513788957970691722144377...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258990 (3,4), A258991 (4,4).

RealDigits[(25/12)*Zeta[6] - Zeta[3]^2, 10, 103] // First

zetamult([2,4]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(2,4) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^4)) = (25/12)*zeta(6) - zeta(3)^2.

Equals Sum_{i, j >= 1} 1/(i^4*j^2*binomial(i+j, i)). - Peter Bala, Aug 05 2025

A258984 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.088483382454368714294327839085760456647978752386750591674889276559474...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Join[{0}, RealDigits[Zeta[3]^2 - (4/3)*Zeta[6], 10, 107] // First]

zetamult([4,2]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(4,2) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^4*n^2)) = zeta(3)^2 - (4/3)*zeta(6).

A258985 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.03857512434275325550592546437299557001973484169890900833104937293358...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Join[{0}, RealDigits[5*Zeta[2]*Zeta[5] + 2*Zeta[3]*Zeta[4] - 11*Zeta[7], 10, 104] // First]

zetamult([5,2]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(5,2) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^5*n^2)) = 5*zeta(2)*zeta(5) + 2*zeta(3)*zeta(4) - 11*zeta(7).

A258986 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.711566197550572432096973806086402612092561204438339236492222496457686...

Dominique Manchon, Arborified multiple zeta values, arXiv:1603.01498 [math.CO], 2016.
Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Cf. A013663 (zeta(5)), A183699 (zeta(2)*zeta(3)).

RealDigits[(9/2)*Zeta[5] - 2*Zeta[2]*Zeta[3], 10, 103] // First

zetamult([2,3]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(2,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^3)) = (9/2)*zeta(5) - 2*zeta(2)*zeta(3).

Equals Sum_{i, j >= 1} 1/(i^3*j^2*binomial(i+j, i)). More generally, for n >= 2, Sum_{i, j >= 1} 1/(i^n*j^2*binomial(i+j, i)) = zeta(2)*zeta(n) - zeta(n+2) - zeta(n,2). - Peter Bala, Aug 05 2025

A258988 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,3).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.0851598225348336514068060188723673459573395085868773204671034320533...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).

Join[{0}, RealDigits[17*Zeta[7] - 10*Zeta[2]*Zeta[5], 10, 104] // First]

zetamult([4,3]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(4,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^3)) = 17*zeta(7) - 10*zeta(2)*zeta(5).

A258991 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,4).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.08367311301649536161489043654238770543824673255415416836075918355438...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4).

Join[{0}, RealDigits[(1/2)*(Zeta[4]^2 - Zeta[8]), 10, 106] // First]

zetamult([4,4]) \\ Charles R Greathouse IV, Jan 21 2016

(zeta(4)^2-zeta(8))/2 \\ Charles R Greathouse IV, Jan 20 2022

zetamult(4,4) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^4*n^4)) = (1/2)*(zeta(4)^2 - zeta(8)).

A381394 Decimal expansion of the multiple zeta value zetamult(8,2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 22 2025

			0.004122469678399832224046956838694208855812627358468569285245...

Richard E. Crandall and Joe P. Buhler, On the evaluation of Euler Sums, Exp. Math. 3 (4) (1994), 275-285, Table 1.
Debra Maître, Mathematica Package HPL.
Eric Weisstein's MathWorld, Multivariate Zeta Function

MZV's zetamult(a,b): A072691 (zetamult(1,1)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4), A381651 (4,1).

RealDigits[N[MZV[{8, 2}], 120], 10, 105, -1][[1]] (* Amiram Eldar, Feb 25 2025 using the HPL Package *)

zetamult([8, 2]) \\ Amiram Eldar, Feb 25 2025

zeta(r,s) = Sum_{1 <= m < n} 1/(m^s n^r).

More terms from Amiram Eldar, Feb 25 2025

Name corrected by Peter Bala, Aug 15 2025

cp's OEIS Frontend

A258987 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A258983 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A258989 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,4).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A258984 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A258985 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A258986 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A258988 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula