Given a finite set d of distinct positive integers, the restricted partition function W(d,n) counts the number of ways of writing a positive integer n as a sum of elements from d, disregarding order of the summands.

Sylvester showed that for a fixed d, W(d,n) can be written as a finite sum of functions he called "waves", the most basic and important one being a polynomial. I will present an explicit formula for this polynomial part, also known as the first Sylvester wave. This is achieved by way of some identities for higher-order Bernoulli polynomials, one of which is analogous to Raabe's well-known multiplication formula for the ordinary Bernoulli polynomials. As a consequence of the main result we obtain an asymptotic expression of the first Sylvester wave as the coefficients of the restricted partition grow arbitrarily large. As part of this talk I will introduce and discuss a symbolic notation for Bernoulli numbers and polynomials.

(Joint work with Christophe Vignat).