How to determine the value of $\displaystyle f(x) =...
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.Yes, I am aware there is no reason to believe a random power...
View ArticleGeneration of Matrix Algebras without Identity
Past research on matrix subalgebras generated by a given set of generators has one major restriction: most of the literature on the generation of algebras automatically presumes unitality of the...
View Articlegenerating function backtransformation
i have been working with generating functions and i am fed up with always doing the partial fraction decomposition. i have found the following trick that can (hopefully) skippfd.$$\frac{z^j}{(1-z)^k}...
View ArticleProve that $\displaystyle (r+1) \cdot a_{r+1} = (n-r) \cdot a_{r} + (2n+1-r)...
Let $\displaystyle (1+x+x^2)^n = \sum_{i=0}^{2n} a_i x^i$.Then prove that$\displaystyle (r+1) \cdot a_{r+1} = (n-r) \cdot a_{r} + (2n+1-r) \cdot a_{r-1}$What I try : $\displaystyle...
View ArticleCounting the number of labelled unicyclic graphs via Lagrange inversion
I am struggling with the following exercise:Let $\mathcal{U}$ be the class of labelled unicyclic graphs, i.e. connected graphs with precisely one cylce. Express $\mathcal{U}$ in terms of basic...
View ArticleOGF to EGF transformation
From: https://en.wikipedia.org/wiki/Generating_function_transformation#OGF_⟷_EGF_conversion_formulas,It states that if you have a an ordinary generating function of the form...
View ArticleGenerating Function for Bivariate Legendre Polynomials?
I am aware of the following standard generating function for single-variable Legendre Polynomials:$$\sum\limits_{n=0}^{\infty}P_n(x)z^n = \frac{1}{\sqrt{1-2xz+z^2}}$$for $x \in \mathbb{R}, z \in...
View ArticleShuffled image of set
A permutation $\psi: S \rightarrow S$, where $S = \{1,2,\dots,n\}$, is considered $\textit{descriptive}$ if for every $k < n$, the image under $\psi$ of $\{1,2,\dots,k\}$ is not simply...
View ArticleSimplify in closed-form $\sum_n P_n(0)^2 r^n P_n(\cos \theta)$
Simplify in a closed form the sum $$S(\theta)=\sum_{n=0}^{\infty} P_n(0)^2 r^n P_n(\cos \theta)$$ where $P_n(x)$ is the Legendre polynomial and $0<r<1$. Note that $P_n(0)= 0$ for odd $n$ and...
View ArticleCombinatorial proof of identity involving central binomial coefficients $...
The identity I am interested in reads,$$\sum_{k=0}^{n-1} {2k \choose k} \cdot 2^{2(n-k)} = 2n \cdot {2n \choose n}.$$It is not hard to prove using the generating functions, but it seems that there...
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(combinatorics) Smirnov words with probability - frequency problem
I can't understand the red box above, even though it is quite intuitive.When it comes to the average number(frequency) of jth letter in a n-sized Smirnov word,I think $$0\frac{n_0}{N} + 1\frac{n_1}{N}...
View Articlefind the coefficient $a_n$ of a power series
I have the power series$$f(z)=\frac{2}{(1-3z)^{\frac{2}{3}}}+e^{\frac{1}{2}z^2},$$and I am supposed to find an explicit expression for the coefficient of the corresponding sum representation of the...
View ArticleGenerating function of sequence depending on another sequence
Let $a(n,k) = |${$A ⊂ [n]: |A| = k, A$ does not contain two consecutive elements$}|$Prove that $a(n,k) = a(n−1,k)+a(n−2,k −1)$ for $k ≥ 2$and use it to compute the generating function $A_k(x) = \sum_{n...
View ArticleSolving recurrence relation to find a generating function
Say I have the recurrence:$a_t(u,v) = (a_t(u-1,v-1) + a_t(u-1,v)) q^{u-2v+1}$with initial values:$a_t(u,v) = 0$ for $u < 0 $ or $ L < v < 0 $Where $ L = floor((u-t-1)/2) $ and $a_t(0,0) =...
View ArticlePartitions without repetition
I want to know how many partitions without repetition 19 has. I know I should see the coefficient of $x^{19}$ in $$\prod_{k=1}^\infty(1-x^k),$$but i'm having trouble finding it. Ay hint?
View ArticleThinking of binomial coefficient of $(x+1)^n$ in terms of generating function...
The coefficient of $x^r$ in the expansion of $(x+1)^n$ is $\binom{n}{r}$ which combinatorially represents the number of ways to choose $r$ objects from $n$ objects, but in terms of generating function,...
View ArticleCounting with coefficient of an expression
In an examination, the score in each of the four languages – English, Math, Science and Arts- can be integers between $0$ and $10$. Then what is the number of ways in which a student can secure a total...
View ArticleWhy does this generating function not work - where is the mistake?
I have to solve the recurrent equation:$a_0 = 0, a_1 = 1, a_{n+2} - a_{n+1} + a_n = 0$My approach is quite normal. So let $A(x) = a_0+a_1x+...$ be the generating function and $Q(x) = 1-x+x^2$ be a...
View ArticleExample of a generating polynomial that is not log-concave
Let $[n] = \{1,2,\ldots,n\}$, and let $\mu$ be a probability measure over $2^{[n]}$. In a problem I was working on, I wanted to show that a function of the form is log-concave$$g_\mu(x) =...
View ArticleSpecies of permutations with only even cycle lengths
Let $e(2n)$ be the number of permutations $\sigma $ of a set of $2n$ elements with the property that every cycle of $\sigma $ has even length. Find the exponential generating function $\sum_{n} e(2n)...
View Article$\sum_{n=R}^{\infty}{\binom{n}{R}\frac{a^n}{(1+a)^{n+1}}} = a^R$
for any $R \in \mathbb{N}$ I want to show the identity$$\sum_{n=R}^{\infty}{\binom{n}{R}\frac{a^n}{(1+a)^{n+1}}} = a^R$$I had an argument that I really liked but unfortunately it has some mistakes,...
View ArticleFind generating function for the number of partitions which are not divisible...
I'm trying to find the generating function for the number of partitions into parts, which are not divisible by $3,$ weighted by the sum of the parts. My idea is that we get the following generating...
View Articlegenerating function with conditions
Q)In how many ways can 6 balls be taken out of a bag containing 4 identical blue, 5 identical white and 6 identical green balls, provided that the number of green balls cannot be more than the number...
View ArticleHow to approximate $\frac{(2n+1)}{2^n} \int_0^1 \left(...
Let$$f(x) = \frac{1-x^2+\sqrt{1+2x^2-3x^4}}{2}$$How to approximate the integral$$I_n = (2n+1)\int_0^1 f(x)^n dx?$$Experiments seem to indicate that it is something like $cn^{0.75}+1$ where $c$ is...
View ArticleIs this identity I found playing around with generating function with...
Let $I(n) = \int_{0}^\pi sin^n(x) dx$ , using $sin^2(x) = 1-cos^2(x)$ and integrating by parts we get.$$ \begin{align} I(n) = \dfrac{n-1}{n} I(n-2) \end{align} $$With $I(0) = \pi$ and $I(1) = 2$ now...
View ArticleTaylor series of the Chebyshev polynomials?
How to derive the Taylor expansion of the Chebyshev polynomials$U_n (x)$ at $x=1$?I tried to start from the generating function, but thereis a sum of products of two binomial coefficients I do not know...
View ArticleExtracting colored numbered balls and calculating the probability of their...
Let's say I have an urn with B balls colored in C = C0, C1,..., CC-1 different colors. C > 1.I start with a quantity Q = 1,I extract (without replacement) E balls, one after the other. Depending on...
View ArticleThe generating function for the Fibonacci numbers
Prove that $$1+z+2z^2+3z^3+5z^4+8z^5+13z^6+...=\frac{1}{1-(z+z^2)}$$The coefficients are Fibonacci numbers, i.e., the sequence $\left\{1,1,2,3,5,8,13,21,...\right\}$.
View ArticleWhat's the point of the characteristic function away from zero?
If all I'm interested is in the moments of my random variable $X$, then given its characteristic function $\varphi _{X}(t)$ we have$$\operatorname {E} \left[X^{n}\right]=i^{-n}\left[{\frac...
View Article$a_{m, n}$ is coefficient of $x^n$ in expansion $(1+x+x^2)^m.$ Prove $0\leq...
Problem Statement: Let $a_{m, n}$ denote the coefficient of $x^n$ in the expansion $(1+x+x^2)^m.$ Prove that for all $k\geq 0$,$$0\leq \sum_{i = 0}^{\left\lfloor 2k/3\right\rfloor} (-1)^ia_{k-i,i} \leq...
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