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Pascal's Triangle for Graduate Students A guide to the binomial coefficients for those exploring the far reaches of mathematics

    While Pascal's Triangle itself might seem a quaint presentation of ideas a graduate student has already mastered during a successful undergraduate study program, there are still many secrets held by the array and useful techniques in higher many fields of higher mathematics that come back to use of the binomial coefficients time and time again.  Here are a few highlights of the information presented on this website that may be of use to a graduate student.

Modern Algorithmic Methods: This page is a brief summary of the book "A=B" by Petkovšek, Wilf and Zeilberger, which is available online.  The modern methods for finding closed formulas for difficult summations, including of course summations involving binomial coefficients, can be of use to people studying fields as diverse as differential equations and computer science.

Identities Page:  This list of identities involving the binomial coefficients, while not completely exhaustive, does list many identities the graduate student may not be aware of, including a long list of named identities, like Dixon's Identity, Abel's Identity, the identities from Pascal's original treatise, etc.; also, there are many of the relations between the binomial coefficients and famous sets of numbers like the Fibonaci sequence, the Stirling numbers of the First and Second Kinds and the numbers found in Euler's Triangle.  Most of the identities that have proofs attached to them are proofs of an elementary nature (some so elementary that only a hint is given instead of a proof), but some of these proofs may not be known to every grad student, and their relative simplicity often belies their mathematical power.

Recent Research:  This is a list of papers published over the past few years whose proofs rely on relations between the binomial coefficients.  Knuth once wrote, "There are so many relations present that when someone finds a new identity, there aren't many people who get excited about it anymore, except the discoverer!"  While discovering of the secrets of Pascal's Triangle may not be the forefront of math that it was in the day of Pascal or Tartaglia or Newton, and it might not be enough to secure you a job as it did Professor Moriarity, people are still publishing their new insights into Pascal's Triangle, and every grad student knows that publishing is still the name of the game.