Questions tagged [binomial-coefficients]
For questions that explicitly reference the binomial coefficients, Pascal's Triangle, and Binomial identities.
406
questions
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3
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A closed form (or tight upper bound) for $\sum_{j=0}^{2m} (-1)^j (m-j)^{2m+2k} \binom{2m}{j}$
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=...
5
votes
3
answers
843
views
How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?
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
$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n(...
2
votes
1
answer
223
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An integer sequence related to Pascal’s triangle
We need someone expert in binomial coefficients (subject 11B65) to recognize the integer sequence generated by an iterative formula we have encountered while working on a project about Pascal’s ...
4
votes
1
answer
222
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About the exact origin of a binomial congruence
Given a prime $p$ and an integer $0 \leq k \leq p-1$, a famous congruence on binomial coefficients states:
$$\binom{p-1}{k} \equiv (-1)^k \pmod{p}$$
It is generally taught as a consequence of Pascal’s ...
0
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2
answers
126
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Asymptotic bound of a simple alternating binomial sum
I'm a rather inexperienced researcher, I've been stuck on a question for a while. I would like to find the largest $N = f(n)$ that satisfies the following inequality:
$$\sum_{j = 0} ^ n p^{n - j} (-1)...
2
votes
5
answers
904
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Binomial 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 ...
3
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1
answer
632
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binomial 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 ...
8
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0
answers
431
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Does 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 $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...
3
votes
1
answer
415
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Identities 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 (-1)^{k-i}\...
24
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2
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Are (55, 165, 495, 1485) and (286, 1716, 10296, 61776) the only geometric sequences of length 4 among non-trivial binomials?
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 ...
10
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1
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Solve $\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 ...
0
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1
answer
168
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Question in a paper by Erdős on divisibility properties of central binomial coefficient
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 (...
13
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1
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420
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Four new series for $\pi$ and related identities involving harmonic numbers
Recently, I discovered the following four new (conjectural) series for $\pi$:
\begin{align}\sum_{k=1}^\infty\frac{(5k^2-4k+1)8^k\binom{3k}k}{k(3k-1)(3k-2)\binom{2k}k\binom{4k}{2k}}&=\frac{3\pi}2,\...
8
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Why 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=...
1
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1
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295
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sum 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} .
$$
Michael Lugo suggest a way with geometric series ...
2
votes
2
answers
198
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Sign-reversing involution for $q$-binomial coefficient at $q=-1$
Consider the q-binomial coefficient $\binom{n}{k}_q$.
One combinatorial way to define it is as follows. Let $W_{n,k}$ be the set of binary words of length $n$ with $(n-k)$ 0's and $k$ 1's. An ...
6
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1
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A summation involving fraction of binomial coefficients
I need to prove the following statement.
Let $ n, g, m, a ,t$ be integers. Prove that the following statement is true for all $ n \geq g(1+2m)+1 $, $ g\geq 2t $, $ m\geq t $, $ 0\leq a <t $, and $ ...
8
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3
answers
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Alternating Sum Involving Catalan Numbers
I was wondering if anyone knew how to obtain a simpler closed form of the following sum(or had any other insights regarding it):
$$\sum_{k=0}^n (-1)^k{n \choose k} C_{2n-2-k} $$
Here $C_n = \frac{1}{n+...
0
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0
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78
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Solving a Catalan-like recursion of polynomials, related to the KdV energies
I am working on a PDE problem. The goal is to connect the higher order energies of the Gross-Pitaevskii equation to those of the Korteweg-de-Vries equation. As these higher order energies are ...
1
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0
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86
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On level-$12$ of the McKay-Thompson series of the Monster and the Domb numbers
(This continues from level 10.) Given some moonshine functions $j_{N}$. There are nice descending and consistent relations for levels $6m$ with $m= 2,3,5,$
$$j_{12A} = \left(\sqrt{j_{12H}} + \frac{\...
1
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0
answers
96
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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 ...
1
vote
2
answers
164
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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 $i=1,2$.
0
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0
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225
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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):
$$
\sum_{i=0}^{n-1}\binom{n+1-i}{...
1
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0
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3D generalization of Gaussian q-binomial coefficient
It is known that the coefficient of $q^t$ in Gaussian binomial coefficient $\binom{m+n}m_q$ equals the number of permutations of the multiset $\{0^m, 1^n\}$ with $t$ inversions.
Is there a closed ...
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New method to compute square roots [closed]
In 2011 when I was in school I created a formula to calculate square roots... For $x\in\mathbb{R}$ with $x>0$ the following holds:
$$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...
3
votes
1
answer
382
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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
$$\sum_{i+j+k=p,\text{ } i,j,k\gt 0}{p\choose ...
0
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0
answers
257
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An alternating sum involving a product of binomial coefficients
I encountered the sum below, where $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$ and $d$ are some given positive constants. Does anyone have an idea how to simplify it?
$$
\sum\limits_{k=1}^{d} \frac{(-1)^{k-1}k}...
4
votes
0
answers
132
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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 $U_n$.
...
2
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0
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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
$$a(n) = a(n-1) + (n-1)a(n-2), a(...
0
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2
answers
231
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Integer solutions of system of inequalities
I am trying to solve a problem in combinatorics and I came up with the following system of inequalities:
$0\leq x<y<z\leq n$ and $x+y<n$ and I am trying to count the number of integer ...
7
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1
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271
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A reference for a sum found in Gould's Combinatorial Identities book
On p. 49 in Gould's book Combinatorial Identities, the author states that the sum $$\sum_{k=0}^{n-1}(-1)^k\binom{n}{k}\binom{2n}{2k}^{-1}$$ "... arises naturally in a statistical problem; it ...
1
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0
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Maybe with the multinomial theorem
I'm looking for some idea to prove
\begin{equation*}
\sum_{\substack{ \left( s_{0},\ldots ,s_{r-1}\right) \in \left\{ \mathbb{N}%
\cup \left\{ 0\right\} \right\} ^{r}, \\ \sum_{a=0}^{r-1}\left( a+1\...
3
votes
1
answer
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Why is this alternating sum involving Catalan numbers $\sum_{i=0}^{\lfloor t/2 \rfloor} (-1)^{i+1} \binom{t-i}{i} C_{t-i-1} = 0$ for all $t$?
I need the result that for all $t$,
$$\sum_{i=0}^{\lfloor t/2 \rfloor} (-1)^{i+1} \binom{t-i}{i} C_{t-i-1} = 0,$$
where $C_{t-i-1}$ is the $(t-i-1)$-th Catalan number. I've checked for $t$ up to ...
13
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1
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565
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A congruence for a product of binomial coefficients?
For every prime $p\geq 5$ one seems to have the congruence
$$(-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}\ .$$
(I have checked all primes up to $5000$...
4
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0
answers
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Greatest common divisors of some binomial coefficients
This is cross-posted from math.stackexchange.
While making some computation, I stumbled upon a curious relation among some binomial coefficients.
Consider the sequence of binomial coefficients $a(k,n)$...
4
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2
answers
233
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(Conceptual) proof and/or interpretation of a $q$-binomial identity
There is a $q$-binomial identity that I encountered in one paper I am reading (https://arxiv.org/abs/1910.06193) which probably admits a very simple proof that I do not see: for two nonnegative ...
9
votes
1
answer
387
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Series for $\frac{\log m}{\pi}$ with summands involving harmonic numbers
The classical rational Ramanujan-type series for $1/\pi$ have the following four forms:
\begin{align}\sum_{k=0}^\infty(ak+b)\frac{\binom{2k}k^3}{m^k}&=\frac{c}{\pi},\label{1}\tag{1}
\\\sum_{k=0}^\...
3
votes
2
answers
495
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Binomial coefficient congruence modulo $p^n$
I am interested in the following congruence
$$\binom{ap^n}{bp^n}\equiv \binom{a}{b}\pmod{p^n}$$
I am aware that by some reference in a book the above it should actually hold modulo $p^{3n}$; the ...
4
votes
0
answers
271
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What is the exact value of the series $\sum_{k=0}^\infty \binom{2k}k^4/256^k$?
By Stirling's formula $n!\sim\sqrt{2\pi n}(n/e)^n$, we have
$$\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}^\...
1
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0
answers
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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 ...
1
vote
1
answer
310
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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 ...
4
votes
1
answer
451
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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 {...
1
vote
0
answers
144
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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}{...
7
votes
1
answer
161
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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
$${{\left\langle\matrix {n \cr k}\right\rangle}_r=\frac{\langle n \rangle_r!}{\langle k \rangle_r! \...
5
votes
3
answers
385
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How to calculate the sum of general type $\sum_{k=0}^n {n\choose k} {n\choose k+a} {2 k- n + a \choose r }$?
QUESTION. How to calculate the sum of such general type?
$$\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
2k
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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
$${xy+2\choose 3}={x+2\choose 3}{y+2\choose 3}+4{x+...
0
votes
1
answer
141
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Number of ways to write a finite set of cardinality n as the union of r distinct binary subsets [closed]
I want to know the number of ways to write a finite set of cardinality $n$ as the union of $r$ distinct two-element subsets. Is there a nice formula in binomial coefficients?
1
vote
1
answer
156
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Counting spanning trees of $K_{b+1,w+1}$ with certain properties or calculating a combinatorial sum
For $b,w \geq 0$ let $K_{b+1,w+1}$ be the complete bipartite graph with vertices $a_1,...,a_{b+1}$ on the left hand side and $c_1,...,c_{w+1}$ on the right hand side. For given $1 \leq d \leq w$ and $...
2
votes
0
answers
206
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Question on globally convergent formulas for the Riemann zeta function $\zeta(s)$
Consider the following two formulas for $\zeta(s)$
$$\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)^...
20
votes
3
answers
3k
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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
...