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Combinatorial Interpretations of Convolutions of the Catalan Numbers

Abstract We reintroduce an interpretation of the kth-fold self convolution of the Catalan numbers by showing that they count the number of words in symbols X and Y, where the total number of Y's is k more than the total number of X's, and at no time are there more Y's than k plus the number of X's. Using this, we exhibit some of the wide variety of combinatorial interpretations of the kth-fold self convolution of the Catalan numbers. Finally, we show how these numbers appear as the last column in a truncated Pascal's triangle.