Multiple zeta values are the numbers defined by the convergent series , where , , , are positive integers with . For , let be the sum of all multiple zeta values with even arguments whose weight is and whose depth is . The well-known result was extended to and by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbers and then gave a direct formula for for arbitrary . In this paper we apply a technique introduced by Granville to present an algorithm to calculate and prove that the direct formula can also be deduced from Eisenstein's double product.
"Algorithms for Some Euler-Type Identities for Multiple Zeta Values." J. Appl. Math. 2013 1 - 7, 2013. https://doi.org/10.1155/2013/802791