Riordan group involutions and the ΔΔ-sequence

Several important combinatorial arrays, after inserting some minus signs, turn out to be involutions when considered as lower triangular matrices. Among these are the Pascal, RNA, and directed animal matrices. These examples and many others are in the Bell subgroup of the Riordan group. We characterize all such pseudo-involutions by means of a single sequence called the ΔΔ-sequence. Finally we compute the ΔΔ-sequences for the powers of a pseudo-involution in the Bell subgroup.