Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.2

From Fibonacci to Robbins: Series Reversion and Hankel Transforms

Paul Barry
School of Science
Waterford Institute of Technology


The Robbins numbers An 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.

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(Concerned with sequences A000045 A000108 A000364 A005130 A005156 A007318 A051255 A052547 A052975 A055209 A077998 A121449 A292865.)

Received April 4 2021; revised version received November 17 2021. Published in Journal of Integer Sequences, November 19 2021.

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