- #1
adil
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Does anybody know how to find the coefficient of x^n in 1/((1-x)(1-x^2)(1-x^3))? I've simplified the above to (1+x+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...), but don't know where to go from here. Any ideas?
mathman said:Let n=j+2k+3m, where j,k,m are non-negative integers. Count how many different combinations of (j,k,m) you can get. That's your answer.
adil said:Thanks, but doesn't this method only work for fixed n?... let n=2, then I can get x^2 from x^2.1.1 or 1.x^2.1, so the coefficient would be 2. The method doesn't work for general n does it?
**Chuckle**matt grime said:If it works for every "fixed n" then surely that is a "general" result?
A generating function is a mathematical tool used to represent a sequence of numbers or coefficients as a power series. It is often used in combinatorics and number theory to solve counting problems.
The coefficient of x^n in a generating function represents the number of ways to form a given object with n elements. It is often denoted as [x^n] and is found by multiplying the term with x^n by the appropriate power of x.
To find the coefficient of x^n in a given generating function, you can use the method of partial fractions. This involves breaking down the generating function into simpler functions and then using the formula [x^n] of a power series to find the coefficient.
The generating function for 1/((1-x)(1-x^2)(1-x^3)) is 1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + 11x^6 + 15x^7 + ... This can be found by using the method mentioned in the previous question.
Generating functions can be applied in various fields such as physics, engineering, and computer science, to name a few. They can be used to solve problems related to counting, probability, and optimization. For example, they can be used to find the number of ways to arrange objects or to calculate the probability of a certain outcome in a game.