A381651 -id:A381651 - cp's OEIS frontend

A381653 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,2,2).

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

Offset: 0

Artur Jasinski, Mar 03 2025

			0.1907518241220842136964721118357975989

Index to constants which are multiple zeta values (2,2,2)

Cf. A381651, A381652.

Mathematica
```
RealDigits[3 Zeta[6]/16, 10, 105][[1]]
```
PARI
```
zetamult([2,2,2])
```

Equals 3*A013664/16.

A381652 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,1,3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 03 2025

			0.618349560571269307895639260851864...

Index to constants which are multiple zeta values (2,1,3)

Cf. A381651, A381653.

RealDigits[Zeta[3]^2 - 13 Zeta[6]/16, 10, 105][[1]]

PARI
```
zetamult([2,1,3])
```

Equals zeta(3)^2 - 13*zeta(6)/16.

A381654 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3,1) = zetamult(3,1,2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 05 2025

			0.0792213975652071659990328100778...

Cf. A381651, A381651, A381653.

kk = RealDigits[53 Zeta[6]/24 - 3 Zeta[3]^2/2, 10, 105][[1]]; Prepend[kk, 0]

PARI
```
zetamult([3, 1, 2])
```

Equals 53*zeta(6)/24 - 3*zeta(3)^2/2.

A381655 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,1) = zetamult(3, 1, 1, 1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 11 2025

For complete list of formulas of the positive multiple zeta values up to weight 6 see A381651.

			0.04053689727151973782904590793969648...

Cf. A381651, A381652, A381653, A381654, A381656, A381657.

RealDigits[3*Zeta[6]/4 - Zeta[3]^2/2, 10, 106][[1]]

PARI
```
zetamult([3,1,1,1])
```

zetamult(5,1) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^5*n)) = 3*zeta(6)/4 - zeta(3)^2/2 = zetamult(3,1,1,1).

A381656 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2,1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 11 2025

			0.0323090289916698816984064916802...

Cf. A381651, A381652, A381653, A381654, A381655, A381657

RealDigits[-203 Zeta[6]/48 + 3 Zeta[3]^2, 10, 106][[1]]

PARI
```
zetamult([3,2,1])
```

Equals -203*zeta(6)/48 + 3*zeta(3)^2.

A381657 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4, 1, 1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 19 2025

			0.0174898531690114044259344452675...

Artur Jasinski, Complete list of formulas of the positive multiple zeta values up to weight 6 (2025)
Index to constants which are multiple zeta values (4,1,1)

Cf. A381651, A381652, A381653, A381654, A381655, A381656.

RealDigits[23 Zeta[6]/16 - Zeta[3]^2, 10, 106][[1]]

PARI
```
zetamult([4,1,1])
```

Equals 23*A013664/16 - A002117^2.

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

A381653 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,2,2).

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A381652 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,1,3).

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A381654 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3,1) = zetamult(3,1,2).

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A381655 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,1) = zetamult(3, 1, 1, 1).

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A381656 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2,1).

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A381657 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4, 1, 1).

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

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