Some sequences of integers (Q1825870)

The author describes ``a kind of experimental mathematics'' based on two transformations of sequences and a hope that not seldomly new insights are gained when finding instances where a known sequence is transformed into another. The two transformations of a sequence \(x=(x_ n)\) of non- negative integers are defined through operators S and A, where \(Sx=(y_ n)\). \(Ax=(z_ n)\) and \[ 1+\sum_{n\geq 1}y_ nt^ n=\prod_{n\geq 1}(1-t^ n)^{-x_ n} \] \[ 1+\sum_{n\geq 1}z_ nt^ n=(1- \sum_{n\geq 1}x_ nt^ n)^{-1}. \] A number of remarkable observations concerning applications of these two transformations are included in the paper.