Pascal Eigenspaces and Invariant Sequences of the First or Second Kind

An infinite real sequence $\{a_n\}$ is called an invariant sequence of the first (resp., second) kind if $a_n=\sum_{k=0}^n {n \choose k} (-1)^k a_k$ (resp., $a_n=\sum_{k=n}^{\infty} {k \choose n} (-1)^k a_k$). We review and investigate invariant sequences of the first and second kinds, and study their relationships using similarities of Pascal-type matrices and their eigenspaces.