Category Archives: Stirling numbers

A Proof of Dobinski’s Formula for Bell Numbers

Dobinski’s formula entails the following infinite series expression for the nth Bell number : In this post we’ll work through a proof of Dobinski’s formula.  We’ll need four formulas: The Maclaurin series for : . The formula for converting normal powers to falling … Continue reading

Posted in Bell numbers, sequences and series, Stirling numbers | 2 Comments

Combinatorial Proofs for the Binomial and Alternating Binomial Power Sums

In this post I give combinatorial proofs of formulas for the binomial and alternating binomial power sums:  and Here’s the first. Identity 1. . Proof.  Both sides count the number of functions from  to subsets of .  For the left … Continue reading

Posted in binomial coefficients, combinatorics, Stirling numbers | 1 Comment

A Combinatorial Proof for the Power Sum

Probably the most common formula for the power sum is the one involving binomial coefficients and Bernoulli numbers , sometimes called Faulhaber’s formula: Historically, this, or a variant of it, was the first general formula for the power sum. There … Continue reading

Posted in binomial coefficients, combinatorics, number theory, Stirling numbers | Leave a comment

A Combinatorial Proof of a Stirling Number Formula

One of the fundamental properties of the (unsigned) Stirling numbers of the first kind is that they can be used to convert from rising factorial powers to ordinary powers via the formula   This post gives a combinatorial proof of this … Continue reading

Posted in combinatorics, Stirling numbers | Leave a comment