Algorithms for Some Euler-Type Identities for Multiple Zeta Values

. . . , s k are positive integers with s 1 > 1. For k ≤ n, let E(2n, k) be the sum of all multiple zeta values with even arguments whose weight is 2n and whose depth is k. The well-known result E(2n, 2) = 3ζ(2n)/4was extended to E(2n, 3) and E(2n, 4) by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbers E(2n, k) and then gave a direct formula for E(2n, k) for arbitrary k ≤ n. In this paper we apply a technique introduced by Granville to present an algorithm to calculate E(2n, k) and prove that the direct formula can also be deduced from Eisenstein’s double product.