  Ramanujan (1916) expressed quotients of certain q-series as polynomials of
Eisenstein series of degree 2, 4, 6 and derived the famous Ramanujan's
differential equations. We continue this research with the variants of
Eisenstein-type series which Hahn (2007) recently introduced. We also prove new
formulas of convolution sums for divisor sum functions as subsequent work of
Cheng- Williams (2004) and Huard-Ou-Spearman-Williams (2002).
