Help with generating function problem

In summary, a generating function is a tool used in mathematics to express a sequence of numbers or functions in a compact and systematic way. It is useful in solving combinatorial and counting problems, as it allows for transformation of complex problems into simpler ones. To construct a generating function, a sequence of numbers or functions is defined and expressed in terms of a variable, typically denoted as "x". There are two types of generating functions: ordinary and exponential, with the former representing discrete sequences and the latter representing sequences of derivatives. Generating functions can also be used to solve recurrence relations by transforming them into algebraic equations.
  • #1
beddytear
5
0
Hi.
I'm really struggling with this generating function problem. Any help would be greatly appreciated.

Question:
Find the generating function for the compositions (c1,c2,c3...,ck) such that for each i, ci is an odd integer at least 2i-1.

Second part of question:

Use the above solution to to find the number of compositions of a positive integer n into k parts (c1,c2...ck) such that for each i, ci is an odd integer at least 2i -1.
 
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  • #2
The "generating function" for a list of numbers is, by definition, the power series having those numbers as coefficients.
 

Related to Help with generating function problem

What is a generating function?

A generating function is a mathematical tool used to express a sequence of numbers or functions in a compact and systematic way. It allows for the manipulation and analysis of the sequence in a simpler form.

Why do we use generating functions?

Generating functions are useful in solving combinatorial and counting problems. They allow us to transform complex problems into simpler ones, making it easier to find closed-form solutions or to perform calculations.

How do I construct a generating function?

To construct a generating function, you need to first define a sequence of numbers or functions. Then, you can express this sequence in terms of a variable, typically denoted as "x". The generating function is then formed by grouping the terms together based on the powers of "x".

What is the difference between ordinary and exponential generating functions?

Ordinary generating functions represent discrete sequences, where the coefficient of each term corresponds to an element in the sequence. Exponential generating functions, on the other hand, represent sequences of derivatives, where the coefficient of each term corresponds to the value of the derivative at a specific point.

How can generating functions be used to solve recurrence relations?

Generating functions can be used to solve recurrence relations by expressing the relation in terms of the generating function. This allows for the transformation of the recurrence relation into an algebraic equation, which can then be solved using techniques such as partial fraction decomposition or Taylor series expansion.

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