If I have understood the question well, you are looking for a combinatorial interpretation of nth-fold self-convolution of a sequence. A quick google serach gave me this result on Catalan sequence. May be you can use similar idea for generic sequences.

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 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.