  In this paper, we explore the relation between the index of reducibility and
the Hilbert coefficients in local rings. Consequently, the main result of this
study provides a characterization of a sequentially Cohen-Macaulay ring in
terms of its Hilbert coefficients for non-parameter ideals. As corollaries to
the main theorem, we obtain characterizations of a Gorenstein/Cohen-Macaulay
ring in terms of its Chern coefficients for parameter ideals.
