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An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the Catalan numbers, C n. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product x 0 · x 1 ·⋯ ...
The generating function F for this transformation is of the third kind, = (,). To find F explicitly, use the equation for its derivative from the table above, =, and substitute the expression for P from equation , expressed in terms of p and Q:
The main article gives examples of generating functions for many sequences. Other examples of generating function variants include Dirichlet generating functions (DGFs), Lambert series, and Newton series. In this article we focus on transformations of generating functions in mathematics and keep a running list of useful transformations and ...
One possible generating function for such partitions, taking k fixed and n variable, is = =. More generally, if T is a set of positive integers then the number of partitions of n, all of whose parts belong to T, has generating function
The probability generating function is an example of a generating function of a sequence: see also formal power series. It is equivalent to, and sometimes called, the z-transform of the probability mass function.
For example, every Dyck word w of length ≥ 2 can be written in a unique way in the form w = Xw 1 Yw 2. with (possibly empty) Dyck words w 1 and w 2. The generating function for the Catalan numbers is defined by = =.
(The article on unrestricted partition functions discusses this type of generating function.) For example, the coefficient of x 5 is +1 because there are two ways to split 5 into an even number of distinct parts (4 + 1 and 3 + 2), but only one way to do so for an odd number of distinct parts (the one-part partition 5).
can be proved by the techniques at Stirling numbers and exponential generating functions#Stirling numbers of the first kind and Binomial coefficient#Ordinary generating functions. The table in section 6.1 of Concrete Mathematics provides a plethora of generalized forms of finite sums involving the Stirling numbers. Several particular finite ...