From fibonacci to robbins: Series reversion and hankel transforms

The Robbins numbers A_n have an important place in the study of plane partitions and in the study of alternating sign matrices. A simple closed formula exists for these numbers, but its derivation entails quite sophisticated machinery. Building on this basis, we study the Robbins numbers from a more elementary standpoint, based on series reversion and Hankel transforms. We show how a transformation pipeline can lead from the Fibonacci numbers to the Robbins numbers. We employ the language of Riordan arrays to carry out many of the transformations of the generating functions that we use. We establish links between the revert transforms under discussion and certain scaled moment sequences of a family of continuous Hahn polynomials. Finally we show that a family of quasi-Fibonacci polynomials of 7th order play a fundamental role in this theory.