A simple combinatorial proof of Shapiro's Catalan convolution

TL;DR

This paper presents a straightforward combinatorial proof of Shapiro's Catalan convolution formula and demonstrates its equivalence to an alternating convolution involving central binomial coefficients.

Contribution

It provides a simple combinatorial proof of Shapiro's Catalan convolution and links it to an alternating convolution of central binomial coefficients.

Findings

Established a simple combinatorial proof of Shapiro's Catalan convolution.

Proved the equivalence between Shapiro's convolution and an alternating convolution of binomial coefficients.

Enhanced understanding of the relationship between Catalan numbers and binomial coefficient convolutions.

Abstract

Shapiro proved an elegant convolution formula involving Catalan numbers of even index. This paper gives a simple combinatorial proof of his formula. In addition, we show that it is equivalent with the alternating convolution formula of central binomial coefficients.