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

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

Author

Jean-François Alcover, Jun 16 2015

Keywords

• nonn
• cons
• easy

Examples

			0.67452391403396814049156060825742993927838436513788957970691722144377...
		

Links

• 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), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258990 (3,4), A258991 (4,4).

Programs

Mathematica

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

PARI

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

Formula

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