the inversion relation. This goal in some respects mirrors the more number theoretic (Lambert series) generating function relation guaranteed by the Möbius inversion formula, which provides that whenever the generating functions for the sequences, and , are related by the Möbius transform given by Similarly, the Euler transform of generating functions for two sequences, and , satisfying the relation is given in the form of where the corresponding inversion formulas between the two sequences is given in the reference. The remainder of the results and examples given in this section sketch some of the more well-known generating function transformations provided by sequences related by inversion formulas (the binomial transform and the Stirling transform), and provides several tables of known inversion relations of various types cited in Riordan's Combinatorial Identities book. In many cases, we omit the corresponding functional equations implied by the inversion relationships between two sequences (this part of the article needs more work). The binomial transform The first inversion relation provided below implicit to the binomial transform is perhaps the simplest of all inversion relations we will consider in this section. For any two sequences, and , related by the inversion formulas we have functional equations between the OGFs and EGFs of these sequences provided by the binomial transform in the forms of and The Stirling transform For any pair of sequences, and , related by the Stirling number inversion formula these inversion relations between the two sequences translate into functional equations between the sequence EGFs given by the Stirling transform as and Tables of inversion pairs from Riordan's book These tables appear in chapters 2 and 3 in Riordan's book providing an introduction to inverse relations with many examples, though which does not stress functional equations between the generating functions of sequences related by these inversion relations. The interested reader is encouraged to pick up a copy of the original book for more details. Several forms of the simplest inverse relations Gould classes of inverse relations The terms, and , in the inversion formulas of the form forming several special cases of Gould classes of inverse relations are given in the next table. For classes 1 and 2, the range on the sum satisfies , and for classes 3 and 4 the bounds on the summation are given by . These terms are also somewhat simplified from their original forms in the table by the identities The