Polynomial forms for alternating sums of products of binomial-Catalan numbers (Q2891261)

The subject of the paper under review is the number sequence given by NEWLINE\[NEWLINEf_n(m) = \sum_{k\geq 0} (-1)^k \left(\begin{matrix} n-k \\ k \end{matrix}\right) C_{n+m-k},\quad n\geq 0.NEWLINE\]NEWLINE When \(m=-1\) we have the sequence NEWLINE\[NEWLINE \sum_{k\geq 0} (-1)^k \left(\begin{matrix} n-k \\ k \end{matrix}\right) C_{n-1-k},\quad n\geq 0,NEWLINE\]NEWLINE so that \(f_n(m)\) is a generalization of the binomial-Catalan sequence. About this function two conjectures are made and proved. Conjecture I is that \(f_n(m)\) satisfies the following relation: NEWLINE\[NEWLINEf_{n+1}(m) - f_n(m) = f_{n+2}(m-1),\quad m,n \geq 0;NEWLINE\]NEWLINE and Conjecture II is that \(f_n(m)\) may be represented as polynomials of degree \(m\) in \(n\).NEWLINENEWLINEAs well, it is shown that the functions \(f_n(m)\) are integer-valued, positive-definite in sign and have \(n+1\) as a common factor for \(n\geq 0\), \(m \geq 1\).