  Extending Eulerian polynomials and Faulhaber's formula 1, we study several
combi-natorial aspects of harmonic sums and polylogarithms at non-positive
multi-indices as well as their structure. Our techniques are based on the
combinatorics of non-commutative generating series in the shuffle Hopf algebras
giving a global process to renormalize the divergent polyzetas at non-positive
multi-indices.
