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

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

Author

Jean-François Alcover, Jun 16 2015

Keywords

• nonn
• cons
• easy

Examples

			0.711566197550572432096973806086402612092561204438339236492222496457686...
		

Links

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

Crossrefs

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)).

Programs

Mathematica

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

PARI

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

Formula

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