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Generalising Pascal's Triangle

Published online by Cambridge University Press:  01 August 2016

Barry Lewis*
Affiliation:
21 Muswell Hill Road, London N10 3JB

Extract

Pascal's triangle, the Binomial expansion and the recurrence relation between its entries are all inextricably linked. In the normal course of events, the Binomial expansion leads to Pascal's Triangle, and thence to the recurrence relation between its entries. In this article we are going to reverse this process to make it possible to explore a particular type of generalisation of such interlinked structures, by generalising the recurrence relation and then exploring the resulting generalised ‘Pascal Triangle’ and ‘Binomial expansion’. Within the spectrum of generalisations considered, we find exactly four of particular significance: those concerned with the Binomial coefficients, the Stirling numbers of both kinds, and a lesser known set of numbers – the Lah numbers. We also examine the combinatorial properties of the entries in these triangles and a prime number divisibility property that they all share. Thereby, we achieve a remarkable synthesis of these different entities.

Type
Articles
Copyright
Copyright © The Mathematical Association 2004

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References

2. Hardy, G. H. and Wright, E. M., The theory of numbers (3rd edn), The Clarendon Press (1954).Google Scholar
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4. Riordan, J., An introduction to combinatorial analysis, Wiley (1958) p. 44.Google Scholar