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- ...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 formul4 KB (659 words) - 12:54, 7 March 2022
- #REDIRECT [[Generating function]]33 bytes (3 words) - 12:35, 6 July 2007
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- ...s competitions and can be approached by a variety of techniques, such as [[generating functions]] or the [[Principle of Inclusion-Exclusion|principle of inclusio1 KB (208 words) - 02:12, 4 October 2020
- * [[Generating function]]4 KB (615 words) - 11:43, 21 May 2021
- ...k Tiefenbruck]] and is supported by the [[San Diego Math Circle]]. [[user:generating | Andy Niedermaier]] was a coach for 2007-2009, and [[user:MCrawford | Math2 KB (378 words) - 16:34, 5 January 2010
- ...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 formul4 KB (659 words) - 12:54, 7 March 2022
- ...function are termwise equal, the series at <math>x = a + b</math> is the [[Generating function#Convolutions|convolution]] of the series at <math>x = a</math> and5 KB (935 words) - 13:11, 20 February 2024
- The [[Sieve of Eratosthenes]] is a relatively simplistic [[algorithm]] for generating a list of the first few prime numbers. It is a method in which the multiple The Sieve of Sundaram is a relatively simplistic [[algorithm]] for generating all odd prime numbers, less than <math>2n+2</math>. It is a method by which6 KB (985 words) - 12:38, 25 February 2024
- ..._i=C_{i-1}+4i</math>, and note that <math>C_0=1</math>. Now we can create generating functions. <math>F(x)=\sum_{i=0}^\infty C_ix^i</math>. Also, <math>G(x)=\7 KB (1,276 words) - 20:51, 6 January 2024
- 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 te8 KB (1,332 words) - 17:37, 17 September 2023
- * [[Generating functions]]910 bytes (77 words) - 16:23, 18 May 2021
- == 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 go17 KB (2,837 words) - 13:34, 4 April 2024
- ...th> can be equal with some value of <math>x</math>). MAA is pretty good at generating smooth combinations, so in this case, the AM-GM works; however, always try4 KB (703 words) - 02:40, 29 December 2023
- == Solution 2 (Generating Functions)==3 KB (515 words) - 04:29, 27 November 2023
- ...for each of the terms, and obtain <math>(x+x^3+x^5\cdots)^4</math> as the generating function for the sum of the <math>4</math> numbers. We seek the <math>x^{985 KB (684 words) - 11:41, 13 August 2023
- ===Solution 6 (generating functions)=== The generating function for this is <math>(x+x^2)</math> since an ant on any vertex of the15 KB (2,406 words) - 23:56, 23 November 2023
- ...+1}</math>, <math>M\geq 0</math> be the length of the longest jump made in generating <math>J_{i_0,k_0}</math>. Such a jump can only be made from a number that i7 KB (1,280 words) - 17:23, 26 March 2016
- == Generating Subset == ...ubset is said to be ''minimal'' if on removing any element it ceases to be generating.3 KB (561 words) - 00:47, 21 March 2009
- * [[Generating functions]]705 bytes (64 words) - 16:22, 18 May 2021
- We use [[generating function]]s to represent the sum of the two dice rolls: <center><math>(x+x^1 KB (210 words) - 01:30, 3 January 2023
- ...n if two squares in the row are shaded, then the row is represented by the generating function <math>ab+ac+ad+bc+bd+cd</math>, which we can write as <math>P(a,b,13 KB (2,328 words) - 00:12, 29 November 2023
- 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 is6 KB (983 words) - 13:42, 8 December 2021
- * [[Generating function]]1 KB (251 words) - 15:13, 11 August 2020
- The [[generating function]] for <math>a, b, c,</math> and <math>d</math> is <math>x+x^2+x^3+1 KB (172 words) - 09:56, 18 June 2008
- ...[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 ==10 KB (1,508 words) - 14:24, 17 September 2017
- ==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 coe6 KB (909 words) - 15:39, 8 August 2022
- ...math> to <math>M(X)</math>, and let <math>(u_{i},v_i)_{i\in I}</math> be a generating set of the equivalence relation <math>R(x,y)</math> defined as <math>f(x) =4 KB (887 words) - 13:19, 6 July 2016
- '''Corollary 4.''' Let <math>X</math> be a generating subset of <math>G</math>. Then <math>D(G)</math> is the normal subgroup ge4 KB (688 words) - 20:11, 28 May 2008
- ...the expansion of the coefficients of the product of two polynomials (or [[generating functions]]).1 KB (190 words) - 00:57, 31 May 2016
- ...s an applied discipline, mathematicians have developed various methods for generating approximate solutions to intractable problems. The utility of such methods2 KB (322 words) - 21:03, 11 February 2009
- ...tangent]] numbers. These latter names are the result of the remarkable [[generating function]] (or equivalently [[Taylor series]]) [[identity]]2 KB (246 words) - 12:50, 6 August 2009
- ..._{n-1} + \cdots + a_kG_{n-k}</math> be a linear recurrence. Consider the [[generating function]] given by19 KB (3,412 words) - 14:57, 21 September 2022
- == Solution 2 (Generating Functions) == ...r horizontally is equally likely, we can write all the possible paths as a generating function:2 KB (321 words) - 08:40, 30 June 2023
- This can be solved quickly and easily with [[generating functions]]. The generating functions for these coins are <math>(1+x)</math>,<math>(1+x)</math>,and <ma3 KB (470 words) - 22:15, 27 August 2023
- ==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^{7 KB (1,147 words) - 21:58, 23 January 2024
- ...the first, second, third respective squares are <math>1</math>'s. Then the generating function representing the possible events that exclude a row of <math>1,1,1 Therefore, the generating function representing the possible grids where no row is filled with <math>6 KB (1,057 words) - 01:58, 8 January 2023
- ...h>a,x,y,k,b.</math> Now we use generating functions to finish. We find the generating function of the whole expression is <math>(x + x^2 + \cdots)^4 \cdot (x^2+x9 KB (1,535 words) - 01:28, 16 January 2023
- Use [[generating functions]]. For each department, there is 1 way to pick 2 males, 4 ways to2 KB (328 words) - 01:26, 28 January 2023
- Consider the generating function for a 12 sided die. When rolled n times, the generating function is <math>(x^1+x^2+\hdots+x^{12})^n</math>. This polynomial is clea949 bytes (159 words) - 03:01, 5 April 2012
- .../math> of two elements, and on the left, we have the factorisation of this generating function that considers the breakdown of any given monic polynomial into mo8 KB (1,348 words) - 09:44, 25 June 2022
- The process of choosing a block can be represented by a generating function. Each choice we make can match the 'plastic medium red circle' in3 KB (507 words) - 20:48, 6 December 2021
- ==Solution 5 (Generating Functions)==11 KB (1,677 words) - 23:54, 4 February 2022
- Recall the generating function of the Catalan numbers,3 KB (528 words) - 10:56, 16 April 2024
- | Generating Functions4 KB (594 words) - 10:21, 31 August 2021
- ==Solution 7 (Generating Functions)==12 KB (2,047 words) - 22:02, 9 January 2024
- ...asing the color number by <math>1,2,3\pmod4</math>, respectively. Thus the generating function that represents going through all six borders is <math>A(x)=(x+x^220 KB (3,328 words) - 21:13, 5 April 2024
- ...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's5 KB (975 words) - 14:32, 30 August 2018
- ...hat have no solutions in <math>S</math> a similar proof holds, but instead generating the terms <math>S_{m+1} = 10N - M</math> and <math>S_{m+2} = -5N + M</math>3 KB (584 words) - 07:56, 16 April 2018
- ...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.18 KB (2,878 words) - 01:47, 16 December 2023
- To solve part (b), we look back at the original method of generating solutions. Define <math>a_n</math> and <math>b_n</math> to be the pair repr10 KB (1,657 words) - 19:08, 24 April 2023
- We can use generating functions, where <math>(x+x^2+...+x^6)</math> is the function for each die.3 KB (498 words) - 20:57, 7 October 2023
- ==Solution 8 (Generating Functions)== ...t or out of the set. Therefore, given <math>n\in U</math>, the probability generating function is26 KB (4,044 words) - 13:58, 24 January 2024
- ==Solution 6(Generating Functions???)== Therefore the generating function for the paths beginning at <math>1</math> or <math>A</math> can be11 KB (1,934 words) - 12:18, 29 March 2024
- ==Solution 7(Generating functions)== Now, using multi-variable generating functions, we get:19 KB (2,942 words) - 21:22, 21 January 2024
- ...as the [https://artofproblemsolving.com/wiki/index.php/Generating_function generating function] ...x <math>Z(t_1,t_2,...,t_N)</math> where <math>|\mathcal{X}|=N</math>. The generating function for the number of ways to paint <math>\mathcal{X}</math> in colors2 KB (308 words) - 19:51, 24 November 2021
- ==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)=7 KB (1,152 words) - 14:13, 29 February 2024
- ==Solution 4 (Generating Function Bash)== Notice <math>\text{lcm}(105,70,42,30) = 210</math>, so we can rewrite the generating function as6 KB (874 words) - 00:00, 20 April 2024
- ==Solution 4 (Generating Function)== Use a generating function, define <math>c_{n}\cdot x^{n}</math> be <math>c_{n}</math> ways f8 KB (1,309 words) - 00:31, 6 January 2023
- ==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 t9 KB (1,513 words) - 22:38, 18 February 2024
- ...t we can generate the number of valid sequences of <math>A</math> by first generating all sequences of <math>b</math> such that <math>b_i\leq{}b_{i+1}</math> for9 KB (1,370 words) - 09:08, 19 February 2024
- == Solution 6 (generating functions) == ...th> bills, <math>\$5</math> bills, and <math>\$10</math> bills. We can use generating functions to find the coefficient of <math>x^{80}</math>:6 KB (945 words) - 15:44, 5 February 2024
- We use the generating functions approach to solve this problem. ==Solution 10 (Generating Functions)==15 KB (2,516 words) - 20:31, 15 January 2024