Questions tagged [binomial-coefficients]
For questions that explicitly reference the binomial coefficients, Pascal's Triangle, and Binomial identities.
27 questions from the last 365 days
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Closed form for $\sum\limits_{k=0}^{n} [\operatorname{wt}(k) = m]$ where $\operatorname{wt}(n)$ is the binary weight of $n$
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$).
Let $a(n,m)$ be the family of integer sequences such that
$$
a(n,m) = \sum\limits_{k=0}^{n} [\operatorname{wt}(...
- 4,938
0
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0
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86
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How to prove the following equation (involving multiple binomial coefficients sum)?
I encountered the equation below, encountered a problem that has been bothering me for a long time
Does anyone have an idea how to prove it? I would be extremely grateful to you if you come up with an ...
- 41
0
votes
1
answer
169
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Partial sums of binomial coefficients and related family of polynomials
Let $a(n)$ be A302117. Here
$$
a(n) = 4(n-1)a(n-1) - \frac{1}{3}\prod\limits_{k=0}^{n-1}(2k-3), \\
a(0) = 0.
$$
Let
$$
T(n,k) = \sum\limits_{i=0}^{k} \binom{n}{i}.
$$
Let $P_n(z)$ be the family of ...
- 4,938
4
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0
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168
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How to prove the following equation (which involves binomials and determinant of 2×2 matrices)?
I have tried many ways to prove the following equation, such as the method of induction and expanding all the terms in the summation,but things got more complicated.I could not find an appropriate ...
- 41
0
votes
2
answers
115
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Upper bounds on quotients of binomial coefficients
Let $\gamma>1$ be a real number and let $n\in \mathbb{N}$.
Define $f\colon\mathbb{N}\to[0,1]$
$$
f(n_0) = \frac{\binom{n-n_0}{m}}{\binom{n}{m}},
$$
where
$$
m = \Big\lfloor{\frac{n}{\lceil\gamma ...
0
votes
1
answer
98
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Only special permutations result in a constant expression when permuting coefficients in a sum involving binomials?
Fix $n\geq 1$ and let $p_k(x) := x^k(x-1)^{n-k}$.
Suppose $\pi$ is a permutation on $\{0,1,\dotsc,n\}$, such that
$$
\sum_{k=0}^n (-1)^k \binom{n}{k} p_{\pi(k)}(x) \text{ is a constant}.
$$
Must it be ...
- 15.8k
16
votes
0
answers
309
views
Randomized Pascal's triangle: What is the average of all the numbers?
This question was posted on MSE. It received some interesting responses, but no definite answer.
Let's build a variation of Pascal's triangle. We write $1$'s going down the sides, as usual. Then for ...
- 3,527
5
votes
0
answers
183
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On the polynomials $\sum_{k=0}^n\binom{n+k}k^m q^k$
A sequence of polynomials
$$P_0(q),\ P_1(q),\ P_2(q),\ \ldots$$
with real coefficients is called $q$-log-convex if for each $n=1,2,3,\ldots$ every coefficient of the polynomial $P_{n+1}(q)P_{n-1}(q)-...
- 15.6k
1
vote
0
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138
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Eisenstein triples (and triangles with rational sides and a rational-degree angle) in Pascal's triangle
This question leads to a follow-up: are there any Eisenstein triples (satisfying $a^2\pm ab+b^2=c^2$) in one row of Pascal's triangle apart from the following:
$\binom{23}{8}^2+\binom{23}{8}\binom{23}{...
- 2,370
22
votes
1
answer
1k
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We have $\binom{62}{26}^2+\binom{62}{27}^2=\binom{62}{28}^2$. How many other Pythagorean triples are contained in a single row of Pascal's triangle?
At MSE I asked, "Does any row of Pascal's triangle contain a Pythagorean triple?" The answer is yes; the example $\binom{62}{26}^2+\binom{62}{27}^2=\binom{62}{28}^2$ was given. In that ...
- 3,527
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0
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Are there any numerically-plausible perfect binary code parameters besides (90,2)? [duplicate]
(Formerly on Math StackExchange here, without much progress.)
In order for a perfect binary code on $n$ symbols to correct $k$ errors, we need the sum
$${n\choose 0}+{n\choose 1}+\ldots+{n\choose k}$$
...
- 3,068
4
votes
2
answers
218
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how to prove identity for nth derivative of $(\text{arctanh}(x))^j$?
this question asked on MSE
I worked on integral problem and I got that
$$ \int_0^1 \frac{x^n}{\ln \left(\frac{1-x}{1+x} \right) } dx=-\frac{2}{(n+1)!}\sum_{j=1}^{n+1}F(n,j) \eta'(-j)$$
where $\eta(x)$ ...
- 513
0
votes
1
answer
170
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Summation of binomial coefficients with alternating signs
For a fixed $\alpha > 1$ and integer $n$, I want to provide some bounds or scaling results for the following summations
$$S_1(n,\alpha) = \sum_{k = 1}^{n} {n \choose k} (-1)^{k + 1} k / (\alpha k + ...
- 25
6
votes
0
answers
752
views
For all $n\in \mathbb{N}$, How to find $\min\{ m+k\}$ such that $ \binom{m}{k}=n$?
I asked this question on MSE here.
Most numbers in pascal triangle appear only once (excluding the duplicates in the same row of the Pascal's triangle) but certain numbers appear multiple times. ...
- 541
4
votes
4
answers
780
views
Bounding a binomial coefficient using the binary entropy function
I'm reading that recent paper on a new bound for diagonal Ramsey and am stuck at the attached "Fact 12.1", which is "standard".
Could anyone please point me to a source for this ...
- 579
1
vote
1
answer
142
views
Asymptotics on sum of product of binomial coefficients
I'm interested in the behavior of the summation
$$S(a,b)=\sum_{k\ge 0}\binom{a-k}{k}\binom{b}{k}.$$
For a fixed $\delta>0$, I would like asymptotic bounds on $S(a,\delta a)$.
With $\delta=1$, this ...
- 541
4
votes
0
answers
207
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Who first considered "Pascal Triangle"? [closed]
Arnold was used saying in his talks,
"Pascal’s triangle, so called, because it was by Chinese discovered"!
How much is he right?
- 85
1
vote
0
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73
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Alternating sum of integer coefficients of the triangles related to Eulerian numbers and binomial transforms
Let $W(n, k, m)$ be an integer coefficients defined for $n > 0, 1 \leqslant k \leqslant n, m > 0$ with $W(n,k,m)=0$ for $n \leqslant 0$ or $k \leqslant 0$ such that
$$
W(n, k, m) = (k+m-1)W(n-1,...
- 4,938
3
votes
1
answer
127
views
Bijective proof of deteminant formula for Hankel matrix of central binomial coefficients
Is there a nice bijective proof of the fact that the determinant of the $(n+1)$-by-$(n+1)$ Hankel matrix whose respective entries are the central binomial coefficients $0 \choose 0$, $2 \choose 1$, $\...
- 19.7k
0
votes
1
answer
158
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Generalized Multinomial Formula
During a computation, the following came up, and I was wondering if there is a generalized multinomial formula which can handle expressions of the following form:
Let $n\in \mathbb{N}_+$ and $w_1,\...
- 5,405
7
votes
1
answer
527
views
Suitable 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
$$
1, 1, 2, 2, 4, 5, 8, 12, 19, 28, 45, 70, 110, ...
- 4,938
5
votes
3
answers
325
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A closed form (or tight upper bound) for $\sum_{j=0}^{2m} (-1)^j (m-j)^{2m+2k} \binom{2m}{j}$
I'm seeking a closed-form expression to the sum
$$ \sum_{j=0}^{2m} (-1)^j (m-j)^{2m+2k} \binom{2m}{j} $$
where for positive integers $m$ and $k$, we know $m \gg k$. Loosely, $k \sim \log(m)$.
When $k=...
- 173
5
votes
3
answers
943
views
How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?
I tried to find the indefinite integral
$$ 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(...
- 513
2
votes
1
answer
236
views
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 ...
- 125
4
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1
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252
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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 ...
- 125
0
votes
2
answers
140
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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
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949
views
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 ...
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