Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

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Coefficient of binomial expansion

The coefficient of $x^3$ is $4$ times the coefficient of $x^2$ in the new expansion of $(1+x)^n$. Find the value of $n$.
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4answers
110 views

calculate-binomio-newton

i am help Calculate: $$(C^{16}_0)-(C^{16}_2)+(C^{16}_4)-(C^{16}_6)+(C^{16}_8)-(C^{16}_{10})+(C^{16}_{12})-(C^{16}_{14})+(C^{16}_{16})$$ PD : use $(1+x)^{16}$ and binomio newton
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2answers
131 views

An interesting property of binomial coefficients that I couldn't prove

So when I was trying to prove the argument in this link I've come up with something. When you extract the left term from the right term, you get the term under them. What is interesting is that as ...
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2answers
71 views

How to derive the Taylor expansion form of a polynomial expression?

I want to change this polynomial into the form $\sum_{k=0}^m a_k x^k$ $$q_m(x)=\sum_{k=0}^m(-1)^k\binom{2m+1}{2k+1}x^k(1-x)^{m-k}$$ I see no way to do this as I fear one might need intricate binomial ...
10
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1answer
86 views

Asymptotic Behavior of a Sum with Binomial Coefficients

The Problem: Find the asymptotic behavior (with respect to $n$) of the following sum $$\sum\limits_{j = 3}^n \binom{n}{j} \frac{(j - 1)!}{2\cdot n^j}. $$ Where the Problem Comes From: If we ...
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2answers
91 views

Algebraic proof that $\sum\limits_{i=0}^n \binom{i}{k} = \binom{n + 1}{k + 1}$

I'm looking for an algebraic proof of this identity for $n, k \in \mathbb{N}$: $$\sum\limits_{i=0}^n \binom{i}{k} = \binom{n + 1}{k + 1}$$ So far, I've turned the left hand side of the equality into ...
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0answers
34 views

Binomial-like distribution

Starting with $1$, for $n$ trials multiply by either $1+p$ or $1-p$, with $0 \le p< \le 1$. Does this distribution have a name? What are its properties, such as density (PDF)? It is like a skewed ...
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1answer
40 views

Equality involving binomial coefficient

I think the following equality is correct, but I'm not sure, so I'm asking you: $$\left(\prod_{\large\tfrac{n}{2}\,<\,p\,\le\,\tfrac{6n}{7}}p\right)\cdot\left(\prod_{\large ...
0
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1answer
46 views

Approximation for a binomial coefficient sequence summation

What is a good approximation to $$\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{\binom{k(k-1)/2}{i}}$$ $$\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{(\log ...
4
votes
0answers
71 views

On a unique(?) binomial property of $3003$

Given the triangular number, $$T_k = \frac{k(k+1)}{2}$$ and remembering that, $$\binom{n}{m}=\binom{n}{n-m}$$ Excluding $a_0=1$, we then have the six-fold (at least) equalities, $$\begin{aligned} ...
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2answers
77 views

Is there a good approximation for this?

What is a good approximation for $\dfrac{k!}{\binom{k^2}{k}}$ as a function of $k$? Is there a $k_0\in\Bbb N$ such that for all $k\gt k_0$, ...
-1
votes
1answer
44 views

Is there a short expression for this?

Is there a closed form expression for $$\Bigg(\binom{n}{k-1}+\binom{n-1}{k-1}+\dots+\binom{k-1}{k-1}\Bigg)(k-1)!= \sum_{i=0}^{n-k+1}\frac{(n-i)!}{(n-k+1-i)!},$$ ...
5
votes
2answers
112 views

When does this sum of combinatorial coefficients equal zero?

$p>2$ is a prime number, $n\in \mathbb{N}$. Is the following statement true or false? Thanks. $$\sum_{i=0}^{\lfloor n/p\rfloor}(-1)^i {n\choose ip}=0$$ iff $n=(2k-1)p$ for some $k\in \mathbb{N}$.
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0answers
20 views

Dividors of binomialcoefficient

Is it true, that $ \prod\limits_{\frac{n}{2}<p\le \frac{6n}{7}} p$ divides $ \binom{3n}{n} $? Thank you in advance. I have no idea how to prove it.
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1answer
27 views

Let $f(n)$ be the base-10 logarithm of the sum of the elements of the $n$th row in Pascal's triangle. [closed]

Let $f(n)$ be the base-$10$ logarithm of the sum of the elements of the $n$-th row in Pascal's triangle. Express $\dfrac{f(n)}{\log 2}$ in terms of $n$. Recall that Pascal's triangle begins $$ ...
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1answer
43 views

Is it possible to evaluate this binomial sum?

Would it be possible to evaluate this sum? $$\sum_{k=0}^{N/2}k\binom{N+1}{k},$$ where $N$ is even? I know that the sum $$\sum_{k=0}^{N+1}k\binom{N+1}{k}=2^N(N+1)$$ (by ...
3
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4answers
54 views

How do i prove that $\sum\limits_{r=0}^k \binom{m}{r}\binom{n}{k-r} = \binom{m+n}{k}$ [duplicate]

I have tried the following: Expanding the coefficients and i end up with something like this: $\sum\limits_{r=0}^k \binom{m}{r}\binom{n}{k-r} = \frac{m!}{(m-r)!r!} \frac{n!}{(n-k+r)!(k-r)!}$ and then ...
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0answers
55 views

How prove $\frac{1}{4a b}\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a}\;<\;\dbinom{a(b+1) }{a}\;<\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a} $

Let $a\in\mathbb N$, and $b\in\mathbb R, b\geq 1$ How prove $\frac{1}{4a b}\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a}\;<\;\dbinom{a(b+1) }{a}\;<\;\left[\frac{(b+1)^{b+1}}{b^{b}}\right]^{a} $
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0answers
69 views

A Summation Challenge

I am trying to understand the solution of problem from its editorial by djdolls' answer,I am not able to understand a particulare step which is as follows: $$S(n)=\sum_0^D (-1)^i \cdot ...
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0answers
23 views

How to calculate the $k$-dimension of a subspace of a polynomial ring?

Let $k$ be an infinite field and $R:=k[x_1,...,x_n]$ the polynomial ring in $n$ indeterminates. Why is the $k$-dimension of $U$ given by $\begin{pmatrix} n+m-1 \\ m\end{pmatrix}$, when $U$ is the ...
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21 views

Sum of series involving factorials [closed]

$$ \sum\limits_{j=0}^{[\frac{n}{l}]}(-1)^{slj}\left( \begin{array}{c} n \\ lj \\ \end{array} \right)^s \frac{x^{kj}}{[(a)_{bj}]^s}, $$ where $l,s,n,k,a,b$ are natural numbers and x is ...
2
votes
2answers
17 views

Generate an integer matrix such that all submatrices are non-singular

I need to generate an $\infty \times N$ integer matrix with a few properties. The top $N$ rows (and $N$ columns) should be the identity matrix. Any square submatrix (meaning the result after ...
0
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3answers
47 views

Lower bound for binomial coefficient

Prove that for sufficiently large $ n $ the following inequality holds: $ \binom{5n}{4n}>12^n $. Thank you in advance.
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1answer
43 views

Evaluating a cube root

How to evaluate $(8.024)^{1/3}$ from $(1+3x)^{1/3}$.I already expand it until $x^3$ but i still can't get the answer. I tried googling for the working using binomial theorem but i failed.
2
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3answers
94 views

How to prove that $(\frac{n}{k})^k\leq{{n}\choose{k}}\leq\frac{n^k}{k!}$?

How to prove that $(\frac{n}{k})^k\leq{{n}\choose{k}}\leq\frac{n^k}{k!}$? I can only manage to see the second inequality, could any one give a hint about the first one?
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Another binomial coefficient sum

In my work I ran across the following binomial coefficient sum: $$ S=\sum_{a=0}^{n-1-l} (-1)^a \binom{n}{l+1+a} \binom{l+a}{l} $$ where $n\geq 0$ and $0\leq l \leq n-1$. I browsed the web and found ...
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42 views

Sum involving binomial $\sum_{k=0}^{n} \binom{3n}{3k}$ [duplicate]

The main question is to evaluate: $$\sum_{k=0}^{n} \binom{3n}{3k}$$ There is a standard technique but I cannot split the sums apart and then add them together. Could you help with this step?
4
votes
3answers
229 views

Nested… binomials coefficients? [closed]

Can I have a proof that this number exists? The number: $$\binom{1}{\binom{2}{\binom{3}{\binom{4}{\vdots}}}}$$ If the number exists, then what is the closed form of that number?
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34 views

Radius of convergence of complex series

I need help for this exercise: We consider the following sequence of function $(f_n)_{n\ge0}$: $$f_n:\mathbb{C} \rightarrow \mathbb{C}$$ $$z \mapsto \frac{1}{p_n}[z(1-z)]^{4^n}$$ where $p_n$ is the ...
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Switching the order of summations of a certain function

I am looking to switch the order of the summations of the following function: $$ \lambda = -\sum_{c=1}^{n-1} \sum_{k=c}^n {k \choose c} \frac{(-1)^k}{k!} f^{k-c}U(-c,k-2c+1,-f)\phi(n,k) $$ I don't ...
2
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0answers
46 views

Finding a closed form for this summation

I have been trying to derive a few identities using some bell polynomials and a technique i have come up with and i came across this summation: $$ \rho(n,k) = \sum_{j=0}^k {k \choose j} {\frac{-j}{2} ...
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1answer
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“Binomiable” numbers

Is there a nice criterion to determine whether a given natural $m$ can be written as a binomial number $\binom{n}{k}$ with $1 < k < n-1$? I've been thinking on this problem with a friend and ...
0
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1answer
18 views

Success rate of a player trying to guess a bitstring with given constriants

For work at my university I try to solve a problem. I have a bit string with given length $len$ and count of active bit $active$ An example could be: 1001 0110 ...
2
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0answers
22 views

Prove for $ \forall n \in \mathbb{N}, \exists x,y,z$ ( $0 \leq x < y < z$ ) such that $ n = \binom{x}{1} + \binom{y}{2} + \binom{z}{3}$ [duplicate]

I'm trying to solve a problem from the combinatorics book. Prove or disprove for $ \forall n \in \mathbb{N}, \exists x,y,z \in \mathbb{N} $ ($0 \leq x < y < z$) such that $$ n = \binom{x}{1} + ...
3
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1answer
64 views

Let n and k be integers such that $n > k ≥ 0$. Show that ${n\choose k }$+ ${n\choose k + 1 }$ = ${n + 1\choose k + 1 }$

I'm trying to prove it using algebra and it didn't get very far. Here is how far I got. Now I know ${n\choose k } = \frac{n!}{k!(n-k)!}$ So the entire expression would be $$\frac{n!}{k!(n-k)!} + ...
2
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2answers
39 views

$\sum_{i=1}^n\frac{1}{i}\binom{n}{i}p^i(1-p)^{n-i}\leq\frac{K}{n} $

How can it be proved that, if $0<p<1$, then $$\sum_{i=1}^n\frac{1}{i}\binom{n}{i}p^i(1-p)^{n-i}\leq\frac{K}{n} $$ for some constant $K$? Thanks in advance for every suggestion.
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Expression for binomial coefficient denominator

I'm trying to find an analytical expression for the denominator of $\pmatrix{-1/2\\k}$ in terms of $k$ when the fraction is fully reduced. E.g., the first several such denominators, starting with ...
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33 views

A complicated summation of binomial coefficients

I am trying to evaluate this sum. I think closed form of this sum is not possible, but there might be some bound or approximate result. So far I was unable to find any approximation. Any help will be ...
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1answer
32 views

How to get value of this binomial coefficient expression?

I am trying to work out an upper bound (big O) of an algorithm I thought of in graph theory field. Basically I have a graph $G=(V,E)$. And a subset of vertices $A=\{a_1,a_2,...,a_k\} ∈ V$ such that ...
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1answer
52 views

Better closed form for generating function $\sum \binom{n}{2k} x^k$

I have a power series $F_n(x) = \sum_k \binom{n}{2k} x^k$, which has a closed form of $F_n = \frac12 \left((1 + \sqrt{x})^n + (1 - \sqrt{x})^n\right)$. $$\begin{align} (1 + \sqrt{x})^n + (1 - ...
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0answers
24 views

Proving that binomial coefficients are integers without induction, combinatorics, or formula for exponent of prime

I know of at least three ways to prove that binomial coefficients are integers. One is combinatorial--binomial coefficients count subsets, and thus are integers. Another is inductive, for example ...
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3answers
148 views

Show that $p \in \left[\frac{4^m}{2\sqrt{m}},\frac{4^m}{\sqrt{2m+1}}\right]$

If the number of ways in which $m$ identical apples can be put in $2m$ boxes, so that no box contains more than one apple, is $p$, prove that $$p \in ...
3
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1answer
44 views

Proof for the upper bound and lower bound for binomial coefficients.

I have seen the bounds $\left(\frac{n}{k}\right)^k \leq {n \choose k} \leq \left( \frac{en}{k}\right)^k$ for integers $n \geq k >0$ for the binomial coefficient. I can prove the upper bound in this ...
2
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4answers
102 views

A combinatorial proof for $\binom mk$+$\binom m{k-1}$=$\binom {m+1}k$

I do realize that there is a elementary proof of this result which follows from applying the formula $$\binom mk=\frac{m \cdot (m-1) \cdot \ldots \cdot (m-k+1)}{k!}.$$ I do wonder if there is an ...
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2answers
37 views

Number of ways to select subsets

In how many ways can two distinct subsets of the set $\text{A}$ of $k$ $(k \geq 3)$ elements be selected so that they have exactly two common elements? I started by choosing two elements (that ...
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2answers
24 views

Mathematical expressions for binomial coefficient and Pochhammer’s Symbol with negative values

I have two questions regarding the binomial coefficient and Pochhammer’s Symbol when they contain negative value; In the following example $\sum\limits_{k=0}^{-n} \binom{-n}{k} \left(a\right)_{-n}$. ...
5
votes
2answers
120 views

Determine the number of subsets

How many distinct subsets of a set $\text{A}$ are there, containing at least $9$ elements, where the total number of elements in set $\text{A}$ is $18$ ? I've solved it by making cases of either ...
2
votes
3answers
44 views

Proving that $i! \mid (p-1)\cdot(p-2)\cdots(p-i+1)$ for $i < p$

Started solving this problem: $$ (a+b)^p \equiv a^p+b^p \pmod{p}$$ where $p\in\mathbb{P}$, $a,b\in\mathbb{Z} $ After a few implications I arrived to this $$ i! \mid ...
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2answers
48 views

Kolmogorov-Zurbenko filter - Calculation of coefficients

I'm currently researching the Kolmogorov-Zurbenko filter and trying to implement it myself as a way to smooth one-dimensional signal strength values. The basic filter per se is pretty easy to ...
3
votes
3answers
197 views

matrix representations and polynomials

I just investigated the following matrix and some of its lower powers: $$M = \left[\begin{array}{cccc} 1&0&0&0\\ 1&1&0&0\\ 1&1&1&0\\ 1&1&1&1 ...