Questions tagged [binomial-coefficients]
For questions that explicitly reference the binomial coefficients, Pascal's Triangle, and Binomial identities.
390
questions
2
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2
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Sign-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 ...
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votes
1
answer
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A 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 $ ...
- 63
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votes
3
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Alternating 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+...
- 449
0
votes
0
answers
72
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Solving 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 ...
0
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0
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76
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On level-$12$ of the McKay-Thompson series of the Monster and the Domb numbers
(This continues from level 10.) Given some moonshine functions $j_{N}$. There are nice descending and consistent relations for levels $6m$ with $m= 2,3,5,$
$$j_{12A} = \left(\sqrt{j_{12H}} + \frac{\...
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1
vote
0
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86
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On level $6$ of the McKay–Thompson series of the Monster and Apéry numbers, et al
After the McKay–Thompson series of levels $1$, $2$, $3$, $4$ of the Monster were mentioned in this MO post, level $6$ has very interesting relations as well. (Level 10 is in this post.)
I. Level-6 ...
- 11.5k
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votes
2
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149
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How fast does this summation grow?
$n,i\in\mathbb N$.
The summation in question is
$$\sum_{k=1}^n\prod_{l=1}^k\binom{2^n}{2^l}^i.$$
How fast does this grow? I am specifically looking at $i=1,2$.
- 13.5k
0
votes
0
answers
180
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Looking for a combinatorial proof of an identity
I've come up with an interesting combinatorial identity (thanks to P. Belmans who precomputed the numbers and pointed out to me that they correspond to OEIS A002697):
$$
\sum_{i=0}^{n-1}\binom{n+1-i}{...
- 1,678
1
vote
0
answers
123
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3D generalization of Gaussian q-binomial coefficient
It is known that the coefficient of $q^t$ in Gaussian binomial coefficient $\binom{m+n}m_q$ equals the number of permutations of the multiset $\{0^m, 1^n\}$ with $t$ inversions.
Is there a closed ...
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1
answer
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New method to compute square roots [closed]
In 2011 when I was in school I created a formula to calculate square roots... For $x\in\mathbb{R}$ with $x>0$ the following holds:
$$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...
- 53
3
votes
1
answer
371
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A combinatorial identity involving binomial coefficients
When I was reading an article by CHUN-GANG JI (A SIMPLE PROOF OF A CURIOUS CONGRUENCE BY ZHAO), he mentioned in the acknowledgement
the following identity
$$\sum_{i+j+k=p,\text{ } i,j,k\gt 0}{p\choose ...
- 33
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0
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An alternating sum involving a product of binomial coefficients
I encountered the sum below, where $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$ and $d$ are some given positive constants. Does anyone have an idea how to simplify it?
$$
\sum\limits_{k=1}^{d} \frac{(-1)^{k-1}k}...
- 109
3
votes
0
answers
123
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Irreducibility of polynomials associated to binomial coefficients
Let $n \geq 2$.
Let $M_n$ be the $(n+1) \times (n+1)$ matrix with entries $\binom{l}{k}$ for $0 \leq l,k \leq n$ and $U_n=M_n + M_n^T$ and let $f_n(x)$ denote the characteristic polynomial of $U_n$.
...
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Integer coefficients such that $T(n,k)=R(n,k)-R(n,k-1)$
Let $a(n)$ be A000085, i.e., the number of self-inverse permutations on $n$ letters, also known as involutions; number of standard Young tableaux with $n$ cells. Here
$$a(n) = a(n-1) + (n-1)a(n-2), a(...
- 2,800
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votes
2
answers
152
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Integer 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 ...
- 1
7
votes
1
answer
233
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A reference for a sum found in Gould's Combinatorial Identities book
On p. 49 in Gould's book Combinatorial Identities, the author states that the sum $$\sum_{k=0}^{n-1}(-1)^k\binom{n}{k}\binom{2n}{2k}^{-1}$$ "... arises naturally in a statistical problem; it ...
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0
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75
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Maybe with the multinomial theorem
I'm looking for some idea to prove
\begin{equation*}
\sum_{\substack{ \left( s_{0},\ldots ,s_{r-1}\right) \in \left\{ \mathbb{N}%
\cup \left\{ 0\right\} \right\} ^{r}, \\ \sum_{a=0}^{r-1}\left( a+1\...
- 11
3
votes
1
answer
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Why is this alternating sum involving Catalan numbers $\sum_{i=0}^{\lfloor t/2 \rfloor} (-1)^{i+1} \binom{t-i}{i} C_{t-i-1} = 0$ for all $t$?
I need the result that for all $t$,
$$\sum_{i=0}^{\lfloor t/2 \rfloor} (-1)^{i+1} \binom{t-i}{i} C_{t-i-1} = 0,$$
where $C_{t-i-1}$ is the $(t-i-1)$-th Catalan number. I've checked for $t$ up to ...
- 33
13
votes
1
answer
540
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A congruence for a product of binomial coefficients?
For every prime $p\geq 5$ one seems to have the congruence
$$(-1)^{(p-1)/2}\prod_{k=0}^{p-1}{p-1\choose k}\equiv 1-p+\frac{3}{2}p^2-\frac{7}{6}p^3\pmod{p^4}\ .$$
(I have checked all primes up to $5000$...
- 16.3k
4
votes
0
answers
91
views
Greatest common divisors of some binomial coefficients
This is cross-posted from math.stackexchange.
While making some computation, I stumbled upon a curious relation among some binomial coefficients.
Consider the sequence of binomial coefficients $a(k,n)$...
- 590
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votes
2
answers
195
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(Conceptual) proof and/or interpretation of a $q$-binomial identity
There is a $q$-binomial identity that I encountered in one paper I am reading (https://arxiv.org/abs/1910.06193) which probably admits a very simple proof that I do not see: for two nonnegative ...
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9
votes
1
answer
312
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Series for $\frac{\log m}{\pi}$ with summands involving harmonic numbers
The classical rational Ramanujan-type series for $1/\pi$ have the following four forms:
\begin{align}\sum_{k=0}^\infty(ak+b)\frac{\binom{2k}k^3}{m^k}&=\frac{c}{\pi},\label{1}\tag{1}
\\\sum_{k=0}^\...
- 13.9k
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votes
2
answers
372
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Binomial coefficient congruence modulo $p^n$
I am interested in the following congruence
$$\binom{ap^n}{bp^n}\equiv \binom{a}{b}\pmod{p^n}$$
I am aware that by some reference in a book the above it should actually hold modulo $p^{3n}$; the ...
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votes
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What is the exact value of the series $\sum_{k=0}^\infty \binom{2k}k^4/256^k$?
By Stirling's formula $n!\sim\sqrt{2\pi n}(n/e)^n$, we have
$$\binom{2k}k\sim\frac{4^k}{\sqrt{k\pi}}$$
and hence
$$\frac{\binom{2k}k^4}{256^k}\sim\frac1{k^2\pi^2}.\tag{1}$$
So the series
$$\sum_{k=0}^\...
- 13.9k
1
vote
0
answers
66
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Closed-form expression for combinatorial summation with a quadratic exponent?
In a current project, I have encountered sums of the form $$A_N(\theta_1,\theta_2) = \sum_{x=0}^{N}{N \choose x} \theta_1^x \theta_2^{x^2}$$ for $\theta_1$ and $\theta_2$ positive reals. My current ...
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1
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1
answer
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How to calculate this limit (if exist)?
I have just asked the calculation of the following summation see here $$S(a,b,m,n_1,n_2)=\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k}, $$
which is motivated by the calculation of the ...
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votes
1
answer
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How to calculate this summation $\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k} $?
Question: How to calculate this summation $S=\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k} $? Where $m<n_1,m<n_2$
Remark1: When $a=b$, I know the above summation $S=a^m\sum_{k=0}^m {...
- 57
1
vote
0
answers
86
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polynomial approximation of hypergeometric function 2F1
I have the following function $T(k_1,k_2)$ resulting from multiphoton transition matrix elements calculations:
$T(k_1,k_2)=\gamma^{-k_2}\sum_{j=0}^{k_1}(j+2)_{l+1}\binom{k_1}{j}(k_1+1)_3(\gamma-1)^{j}{...
- 11
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votes
1
answer
158
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A formula for the generating function of Hoggatt binomials or of some Young tableaux
Let ${\left\langle\matrix {n \cr k}\right\rangle}_r$ denote the $r-$Hoggatt binomials defined by
$${{\left\langle\matrix {n \cr k}\right\rangle}_r=\frac{\langle n \rangle_r!}{\langle k \rangle_r! \...
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3
answers
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How to calculate the sum of general type $\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k- n + a \choose r }$?
QUESTION. How to calculate the sum of such general type?
$$\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k - n + a \choose r }. $$
Some particular examples
$$\sum_{k=0}^n {n\choose k} {n\choose k+a} = ...
29
votes
1
answer
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Reason for breakdown of a nice binomial identity
One has the nice identities
$${xy\choose 1}={x\choose 1}{y\choose 1},$$
$${xy+1\choose 2}={x+1\choose 2}{y+1\choose 2}+{x\choose 2}{y\choose 2}$$
and
$${xy+2\choose 3}={x+2\choose 3}{y+2\choose 3}+4{x+...
- 16.3k
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votes
1
answer
111
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Number of ways to write a finite set of cardinality n as the union of r distinct binary subsets [closed]
I want to know the number of ways to write a finite set of cardinality $n$ as the union of $r$ distinct two-element subsets. Is there a nice formula in binomial coefficients?
- 11
2
votes
1
answer
144
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Counting spanning trees of $K_{b+1,w+1}$ with certain properties or calculating a combinatorial sum
For $b,w \geq 0$ let $K_{b+1,w+1}$ be the complete bipartite graph with vertices $a_1,...,a_{b+1}$ on the left hand side and $c_1,...,c_{w+1}$ on the right hand side. For given $1 \leq d \leq w$ and $...
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0
answers
195
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Question on globally convergent formulas for the Riemann zeta function $\zeta(s)$
Consider the following two formulas for $\zeta(s)$
$$\zeta(s)=\underset{K\to\infty}{\text{lim}}\left(\frac{1}{1-2^{1-s}}\sum\limits_{n=0}^K \frac{1}{2^{n+1}}\sum\limits_{k=0}^n \binom{n}{k} \frac{(-1)^...
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3
answers
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Analogue of Fermat's "little" theorem
Let $p$ be a prime, and consider $$S_p(a)=\sum_{\substack{1\le j\le a-1\\(p-1)\mid j}}\binom{a}{j}\;.$$
I have a rather complicated (15 lines) proof that $S_p(a)\equiv0\pmod{p}$. This must be
...
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0
votes
1
answer
148
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upper bound on sum of product of binomial coefficients
For positive integers $\ell < m < n$, consider a partition of $[2n]$ into two $n$-element sets $(X,Y)$. How many ways are there to choose an $m$-subset $A \subset [2n]$ such that the size of the ...
- 135
3
votes
0
answers
259
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Inequalities for Motzkin polynomials
Let us denote by $M_{n}(t)$ the $n$-th Motzkin polynomial. It is defined by $M_1(t) = M_2(t) = 1$ and
$$ M_{n}(t) = \sum_{i=0}^{\lfloor n/2\rfloor } \frac{1}{n-1-i} \binom{n-1-i}{i} \binom{n-1}{i+1} t^...
- 1,970
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votes
1
answer
321
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Could you please confirm or deny two identities involving weighted Stirling numbers of the second kind?
In the paper [1] below, among other things, Carlitz introduced weighted Stirling numbers of the second kind $R(n,k,r)$. He also proved that the numbers $R(n,k,r)$ can be generated by
\begin{equation*}%...
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0
answers
177
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Counting permutations with a fixed number of descents and an extra condition
I am computing the volumes of certain polytopes and it turns out that knowing a "closed formula" for the following number would help a lot.
Determine the number of permutations $\sigma\in \...
- 1,970
2
votes
0
answers
105
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Divisibility based on central binomial coefficients
For some prime $p$, it is a standard approach based on Kummer's criterion to bound the number of positive integers $n<X$ for some parameter $X$, such that $p\nmid \binom{2n}{n}$. However, if we ...
- 1,032
1
vote
1
answer
338
views
Lower bound and limit of a sum with binomial coefficients
Let
$$A_k = \sum_{i=1}^k i {3k-2i-1 \choose i-1} {2i-2 \choose k-i}$$
$$B_k = \sum_{i=1}^k i {3k-2i-2 \choose i-1} {2i-1 \choose k-i}$$
$$C_k = \sum_{i=1}^k (3k-2i-2) {3k-2i-3 \choose i-1} {2i\...
- 115
5
votes
4
answers
772
views
Limit of a sum with binomial coefficients
Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$
$$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$
$$C_k = \frac{\sum_{i=1}^k(...
- 115
0
votes
1
answer
151
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Restrictions on exponents in multinomial formula
From the multinomial formula we have
$$(x_1 + x_2 + \dotsb + x_m)^n
= \sum_{k_1+k_2+\dotsb+k_m=n, \ k_1, k_2, \dotsc, k_m \geq 0} {n \choose k_1, k_2, \dotsc, k_m}
\prod_{t=1}^m x_t^{k_t}\,.$$
I ...
- 1
1
vote
1
answer
227
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Inequalities between sums of products of certain binomial coefficients
I am a PhD student. During my researches, I often have to deal with inequalities involving sums of binomial coefficients, where the sums are indexed by some set of integer compositions. For example, ...
- 795
3
votes
2
answers
227
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An inequality involving binomial coefficients and the powers of two
I came across the following inequality, which should hold for any integer $k\geq 1$:
$$\sum_{j=0}^{k-1}\frac{(-1)^{j}2^{k-1-j}\binom{k}{j}(k-j)}{2k+1-j}\leq
\frac{1}{3}.$$
I have been struggling with ...
- 115
0
votes
1
answer
185
views
Proof for alternating binomial sum over even powers
I have numerical evidence that
$$ \sum_{k=1}^n (-1)^k\frac{k^{p}}{n+k}\binom{2n-1}{n-k}=0 $$
For $p=2,4,6...2n-2$.
How could this be proved?
- 573
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votes
2
answers
290
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Extended binomial coefficients and the gamma function
For which $(a,b,n) \in \mathbb{Z}^3$ satisfying $a+b=n$ does $\frac{\Gamma(z+1)}{\Gamma(x+1)\Gamma(y+1)}$ approach a limit as $(x,y,z) \rightarrow (a,b,n)$ in $\mathbb{C}^3$, and what is that limit? (...
- 19k
3
votes
0
answers
137
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Flat polynomials with factors of big height
Let $p(x)$ be a polynomial of degree $n$ with all coefficients in $\{-1,0,1\}$ (such polynomials are sometimes called flat). I am wondering how big the coefficients of a factor of $p$ can be. Call ...
- 13k
0
votes
1
answer
184
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A binomial product sum that turns out to be 1
The binomial product sum
\begin{align*}
\sum\limits_{\substack{i_1> i_2> \cdots > i_k\\i_1, i_2, \cdots, i_k \in \{1, 2, \cdots, n-1\}}}\binom{n}{i_1}\binom{i_1}{i_2}\binom{i_2}{i_3}\cdots\...
0
votes
1
answer
246
views
Sum of the first m terms of the expansion $(x+y)^n$
Let $S(x, y, m, n) = \sum\limits_{i=0}^m \binom{n}{i}x^i y^{n-i}$, where $0 < m < n$. I want to derive the relation between $S(x, y, m, n)$ and $S(x, y, m, n-1)$.
Is there any formulas I can use?...
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