...a [[sequence]] which is of interest. Therefore the power series (i.e. the generating function) is <math>c_0 + c_1 x + c_2 x^2 + \cdots </math> and the sequence
Many generating functions can be derived using the [[Geometric sequence#Infinite|sum formul
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4 KB (659 words) - 11:54, 7 March 2022
...[Jacobi theta function]], in particular the [[Jacobi triple product]]. The generating function approach and the theta function approach can be used to study many
== Generating Functions ==
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10 KB (1,508 words) - 13:24, 17 September 2017
== Generating Subset ==
...ubset is said to be ''minimal'' if on removing any element it ceases to be generating.
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3 KB (561 words) - 23:47, 20 March 2009
==Solution 7(Generating functions)==
Now, using multi-variable generating functions, we get:
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20 KB (3,220 words) - 02:24, 14 August 2024
==Solution 6 (Generating Function)==
...<math>1</math> to <math>2</math> would be a change of <math>1.</math> This generating function is equal to <math>(x+x^2+x^3)^4.</math> It is clear that we want t
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10 KB (1,600 words) - 11:28, 12 August 2025
...functions to approach this problem -- specifically, we will show that the generating functions of <math>S(n)</math> and <math>T(n)</math> are equal.
Let us start by finding the generating function of <math>S(n).</math> This function counts the total number of 1's
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5 KB (975 words) - 13:32, 30 August 2018
Alternatively, we can use a [[generating function]] to solve this problem.
The goal is to find the generating function for the number of unique terms in the simplified expression (in te
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9 KB (1,467 words) - 16:43, 31 March 2025
We can apply the concept of generating functions here.
...function for the next 5 games is <math>(1 + x)^{5}</math>. Thus, the total generating function for number of games he wins is
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6 KB (983 words) - 12:42, 8 December 2021
==Solution 8 (Generating Functions)==
...t or out of the set. Therefore, given <math>n\in U</math>, the probability generating function is
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26 KB (4,062 words) - 12:03, 19 January 2025
== Solution 6 (Generating Functions and Roots of Unity Filter / Casework) ==
.../math> states, <math>n</math> steps) is <math>(x+x^2+x^3)^n</math>, so the generating function of interest for this problem is <math>(x+x^2+x^3)^7</math>. Our go
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22 KB (3,562 words) - 00:54, 24 August 2025
==Solution 5: Generating Functions==
We will represent the problem using generating functions. Consider the generating function <cmath>f(x) = (1+x^{1000}+x^{2000}+\cdots+x^{99000})(1+x^{100}+x^{
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7 KB (1,197 words) - 07:07, 15 August 2024
We use the generating functions approach to solve this problem.
==Solution 10 (Generating Functions instead of stars and bars)==
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18 KB (2,899 words) - 16:42, 9 November 2025
===Solution 6 (generating functions)===
The generating function for this is <math>(x+x^2)</math> since an ant on any vertex of the
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15 KB (2,406 words) - 22:56, 23 November 2023
==Solution 3 (Generating Functions)==
We can model this as the generating function <cmath>\left(x^3+x^4+x^6\right)^{10}</cmath> where we want the coe
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6 KB (909 words) - 00:20, 5 January 2025
== Solution 2 (Generating Functions) ==
...r horizontally is equally likely, we can write all the possible paths as a generating function:
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5 KB (708 words) - 00:46, 30 November 2024
...lem using elementary counting methods. This solution proceeds by a cleaner generating function.
...nom{4}{a_1,a_3,a_5,a_7}</math>. We want to add these all up. We proceed by generating functions.
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18 KB (2,901 words) - 12:02, 10 October 2025
==Solution 3 (Generating Functions)==
...>. By expanding the binomials and distributing, <math>f(x,y)</math> is the generating function for different groups of basses and tenors. That is, <cmath>f(x,y)=
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8 KB (1,183 words) - 04:40, 26 July 2024
==Solution 4 (Generating Function Bash)==
Notice <math>\text{lcm}(105,70,42,30) = 210</math>, so we can rewrite the generating function as
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7 KB (1,065 words) - 00:50, 10 May 2024
==Solution 4 (Generating Function)==
Use a generating function, define <math>c_{n}\cdot x^{n}</math> be <math>c_{n}</math> ways f
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10 KB (1,622 words) - 12:02, 21 July 2025
== Solution 5 (generating functions) ==
...bills, <imath>\$5</imath> bills, and <imath>\$10</imath> bills. We can use generating functions to find the coefficient of <imath>x^{80}</imath>:
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9 KB (1,442 words) - 23:11, 11 November 2025
