A symbolic computation technique is used to derive closed-form expressions for an analytic continuation of the Euler–Zagier zeta function evaluated at negative integers. This continuation was recently proposed by Sadaoui. The approach presented here yields explicit contiguity identities, recurrences on the depth of the zeta values and their generating functions. Moreover, it allows to prove that the resulting multiple zeta values computed at negative integers coincide with those obtained by another analytic continuation technique that uses the Euler–MacLaurin summation formula.