Nice question! You are looking for a combinatorial interpretation of nth-fold self-convolution of a sequence. A quick google serach gave me this [result][1] on Catalan sequence. In this paper, a similar result is worked out on a series with Catalan coefficients. May be you can use similar idea for generic sequences. 


  [1]: http://www.emis.de/journals/INTEGERS/papers/l3/l3.pdf