Coefficients in the sum...
Consider a sum $$\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j$$which returns an odd power $n^{2m+1}$ of $n$, for $\ m=0,1,2,...$ given fixed $A_{0,m}, \ A_{1,m}, \ ..., \ A_{m,m}$. The coefficients...
View ArticleSolving a Catalan-like recursion of polynomials, related to the KdV energies
I am working on a PDE problem. The goal is to connect the higher order energies of the Gross-Pitaevskii equation to those of the Korteweg-de-Vries equation. As these higher order energies are...
View ArticleAlternating Sum Involving Catalan Numbers
I was wondering if anyone knew how to obtain a simpler closed form of the following sum(or had any other insights regarding it):$$\sum_{k=0}^n (-1)^k{n \choose k} C_{2n-2-k} $$Here $C_n = \frac{1}{n+1}...
View ArticleA summation involving fraction of binomial coefficients
I need to prove the following statement.Let $ n, g, m, a ,t$ be integers. Prove that the following statement is true for all $ n \geq g(1+2m)+1 $, $ g\geq 2t $, $ m\geq t $, $ 0\leq a <t $, and $...
View ArticleSign-reversing involution for $q$-binomial coefficient at $q=-1$
Consider the q-binomial coefficient $\binom{n}{k}_q$.One combinatorial way to define it is as follows. Let $W_{n,k}$ be the set of binary words of length $n$ with $(n-k)$ 0's and $k$ 1's. An inversion...
View Articlesum of binomial coefficient approximation by geometric series
I follow a subject almost like this link:Sum of 'the first k' binomial coefficients for fixed $N$$$f(N,k) = \sum^{k}_{i=0} \binom{N}{i} .$$Michael Lugo suggest a way with geometric series...
View ArticleBounding a sum of products of binomial coefficients
I am trying to understand the following sums for $k\le n$ :$$\sum_{s=0}^{k} \begin{pmatrix} 2n-s/2\\ s\end{pmatrix}\begin{pmatrix} 2n-3s/2\\ k-s\end{pmatrix}$$$$\sum_{s=0}^{k} \begin{pmatrix} 2n-s\\...
View ArticleWhy does this combinatorial sum vanish?
We define the coefficients $c_{k,k-i}$ of ${n \choose k}$ by the following easy expansion:\begin{align*}& {n \choose k} = \frac{1}{k!} n(n-1) \dots (n-k+1) = \frac{1}{k!} \prod\limits_{t=0}^{k-1}...
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Question in a paper by Erdős on divisibility properties of central binomial...
In Erdős, Graham, Ruzsa, and Straus - On the prime factors of $\binom{2n}n$, at the beginning of the proof of theorem 1, they consider the case where $\log p$ and $\log q$ are commensurable numbers...
View ArticleSolve $\binom{n}{k}=m$ for $(n,k)$
For an integer $m>0$, put $X(m)=\{(n,k):4\leq 2k\leq n \text{ and } \binom{n}{k}=m\}$. Is there an efficient method to calculate $X(m)$? Is there a uniform upper bound for $|X(m)|$?By calculating...
View ArticleCosine of a Partial Sum
Does anyone know of a closed formula for $\displaystyle \cos\left(\sum_{n=0}^m a_n\right)$? I've seen formulas for $\displaystyle \cos\left(\sum_{n=0}^\infty a_n\right)$ and $ \displaystyle...
View ArticleBounding the probability that two binomials are equal
Note: This question was migrated from this earlier post, where it initially appeared. Following suggestions, I moved this into its own question.Let $B_{n,p}$ denote the usual binomial random variable...
View ArticleUpper limit on the central binomial coefficient
What is the tightest upper bound we can establish on the central binomial coefficients $ 2n \choose n$ ?I just tried to proceed a bit, like this:$$ n! > n^{\frac{n}{2}} $$for all $ n>2 $. Thus,$$...
View ArticleIdentities for Bernoulli numbers
I arrived at this formula by inductive reasoning, but I don’t know how to prove it.For any natural numbers $m$ and $k=0,1,2,\ldots, m-1$, $B_i$ - Bernoulli numbers we have:$$\sum_{i=0}^k...
View ArticleDoes the interior of Pascal's triangle contain three consecutive integers?
This question defeated Math SE, so I am posting it here.Consider the interior of Pascal's triangle: the triangle without numbers of the form...
View Articlebinomial coefficients are integers because numerator and denominator form pairs?
I've heard of a claim that when calculating the binomial formula with integer input:$\mathrm{Bin}(n,k):=\prod^k_{i=1}\frac{n+1-i}{i}\in \mathbb{N}\ (\forall n,k\in\mathbb N)$each denominator divides an...
View ArticleInteger solutions of system of inequalities
I am trying to solve a problem in combinatorics and I came up with the following system of inequalities:$0\leq x<y<z\leq n$ and $x+y<n$ and I am trying to count the number of integer...
View ArticleFour new series for $\pi$ and related identities involving harmonic numbers
Recently, I discovered the following four new (conjectural) series for...
View ArticleAre (55, 165, 495, 1485) and (286, 1716, 10296, 61776) the only geometric...
Let's define non-trivial binomial coefficients as values of $\binom{n}{k}$, where $n$ and $k$ are positive integers such that $2 \le k \le \frac{n}{2}$. (Therefore, $6$ is the smallest non-trivial...
View ArticleBinomial series
I am interested in the limit $\frac{\sum_{k=0}^n \sqrt{k}\cdot\binom{n}{k}}{\sqrt{n}\cdot2^n}$ as $n$ goes to infinity. Any reference or argument?In general what do we know about the asymptotic...
View ArticleBinomial supercongruences: is there any reason for them?
One of the recent questions, in factthe answerto it, reminded me about the binomial sequence$$a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2,\qquad n=0,1,2,\dots,$$of the Apéry numbers. The numbers...
View ArticleAsymptotic bound of a simple alternating binomial sum
I'm a rather inexperienced researcher, I've been stuck on a question for a while. I would like to find the largest $N = f(n)$ that satisfies the following inequality:$$\sum_{j = 0} ^ n p^{n - j} (-1)^j...
View ArticleAbout the exact origin of a binomial congruence
Given a prime $p$ and an integer $0 \leq k \leq p-1$, a famous congruence on binomial coefficients states:$$\binom{p-1}{k} \equiv (-1)^k \pmod{p}$$It is generally taught as a consequence of Pascal’s...
View ArticleTwo remarkable weighted sums over binary words
This question builds off of the previous MO question Number of collinear ways to fill a grid.Let $A(m,n)$ denote the set of binary words $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_{m+n-2})$ consisting of...
View ArticleAn integer sequence related to Pascal’s triangle
We need someone expert in binomial coefficients (subject 11B65) to recognize the integer sequence generated by an iterative formula we have encountered while working on a project about Pascal’s...
View ArticleHow to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n...
I tried to find the indefinite integral$$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx)dx$$by using Euler's formula and put $x=\frac{\ln y}{2i}$ I got$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int...
View ArticleA closed form (or tight upper bound) for $\sum_{j=0}^{2m} (-1)^j...
I'm seeking a closed-form expression to the sum$$ \sum_{j=0}^{2m} (-1)^j (m-j)^{2m+2k} \binom{2m}{j} $$where for positive integers $m$ and $k$, we know $m \gg k$. Loosely, $k \sim \log(m)$.When $k=0$,...
View ArticleBinomial coefficient in Andrews' partition book
First of all, I think MathOverflow is a very great community to discuss math, either basic or advanced, and I'm glad to participate here. It's my first post, so I'm sorry if i did anything wrong, and...
View ArticleA divisibility of q-binomial coefficients combinatorially
Let a and b be coprime positive integers. Then the number a+b divides the binomial coefficient ${a+b \choose a}$. I know how to prove this combinatorially - for example after choosing an ordered set of...
View ArticleSuitable closed form for the A079501
Let $a(n)$ be A079501 (i.e., number of compositions of the integer $n$ with strictly smallest part in the first position).The sequence begins with$$1, 1, 2, 2, 4, 5, 8, 12, 19, 28, 45, 70, 110, 173,...
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