Pascal eigenspaces and invariant sequences of the first or second kind

An infinite real sequence {an} is called an invariant sequence of the first (resp., second) kind if an=∑k=0n(nk)(−1)kak (resp., an=∑k=n∞(kn)(−1)kak). We review and investigate invariant sequences of the first and second kind, and study their relationships using similarities of Pascal-type matrices and their eigenspaces.