Article 6YWFK Analogs of binomial coefficients

Analogs of binomial coefficients

by
John
from John D. Cook on (#6YWFK)

There are several numbers that are analogous to binomial coefficients and, at least in Donald Knuth's notation, are written in a style analogous to binomial coefficients. And just as binomial coefficients can be arranged into Pascal's triangle, these numbers can be arranged into similar triangles.

In Pascal's triangle, each entry is the sum of the two above it. Specifically,

pascaltriangle1.svg

The q-binomial coefficients satisfy two similar identities.

pascaltriangle2.svg

Here are the analogous theorems for Stirling numbers of the first

pascaltriangle3.svg

and second

pascaltriangle4.svg

kinds.

And finally, here is the corresponding theorem for Eulerian numbers.

pascaltriangle5.svg

The post Analogs of binomial coefficients first appeared on John D. Cook.
External Content
Source RSS or Atom Feed
Feed Location http://feeds.feedburner.com/TheEndeavour?format=xml
Feed Title John D. Cook
Feed Link https://www.johndcook.com/blog
Reply 0 comments