  We show that the Catalan-Schroeder convolution recurrences and their higher
order generalizations can be solved using Riordan arrays and the Catalan
numbers. We investigate the Hankel transforms of many of the recurrence
solutions, and indicate that Somos $4$ sequences often arise. We exhibit
relations between recurrences, Riordan arrays, elliptic curves and Somos $4$
sequences. We furthermore indicate how one can associate a family of orthogonal
polynomials to a point on an elliptic curve, whose moments are related to
recurrence solutions.
