Summary: Based on the multiple binomial transforms introduced in this paper, the $n$-fold generating and generated sequences of a given integer sequence can be defined and a family of this integer sequence can be constructed. The family sets form a partition of the set of integer sequences. Special attention is paid to the recurrent integer sequences, which are produced by some linear and homogeneous recurrence relations or difference equations. For the recurrent integer sequences, a distinct rule to construct their families is obtained based on the linear difference calculus.

11B99, 11B37

multiple binomial transform, generating sequences, family of integer sequences