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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525 views

$f(x)=\sum_{t}{x \choose t}{n-x \choose k-t}$ - even or odd?

The following function popped in my research: $$f(x)=\sum_{\array{0\le t\le k \\ t\equiv_p a}}{x \choose t}{n-x \choose k-t}$$ Where: n,k are natural numbers and $k\le n$. t is taken over all ...
7
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134 views

Solution of $\large\binom{x}{n}+\binom{y}{n}=\binom{z}{n}$ with $n\geq 3$

I found this question in an old problem set. There's no hint or solution mentioned. For $n \geq 3$, prove or disprove the existence of $(x,y,z) \in \mathbb N^3, ...
6
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131 views

Characterization of Sequences with Integral Binomial Coefficients

For any sequence of positive integers $\{ a_i \}_{i \ge 1}$ we can define the generalized binomial coefficients $\binom{n}{k}_{a}$ as follows: $$m!_a = a_1 a_2 \cdots a_m, \binom{n}{k}_a = ...
6
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153 views

Calculating $\sum_{y=0}^x \Pr[Y= y] \Pr[Z\leq k-y]^2$ when Y,Z are binomially distributed?

Remark: I recently rewrote this post, hoping to get answers! I am analyzing the following experiment: Pick an $x \in \{0,\ldots,2k\}$ uniformly at random Pick $(2k+1)$-bit bitstring $b_1=(u,v_1)$ ...
5
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84 views

Why does this sum of binomial coefficient ratios equal 1?

In the course of doing some calculations comparing unrepeatable sets of event trials, I ended up with the following identity. If my reasoning and my math are correct then this ought to be true, and ...
5
votes
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467 views

Proof for an identity involving a sum of binomial coefficients

I am moving through a On The Average Height of Planted Plane Trees by Knuth, de Bruijn and Rice, 1972), and I would like to trade a weaker result for simpler mathematical tools, because my skills are ...
5
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124 views

Some rare binomial identities

Long ago , I once saw a nontrivial appealing binomial type of identity that I never saw again. It was something along the line of $\Sigma$$\binom{a(x)}{b(y)}$= where $a$ and $b$ where polynomials not ...
5
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187 views

Partial sum over $M$, of ${m+j \choose M} {1-M \choose m+i-M}$?

Is there (likely to be) any formula for $$ \sum_{m' \geq m} {m+j \choose m'} {B - m' \choose m+i-m'} $$ ? I am mainly interested in the case $B=1$ (or a prime power), if that's any ...
4
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87 views

Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
4
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176 views

How to prove that $\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$?

How to prove this: $$\sum_{k=0}^\infty \binom{x}{x-k}\cdot\binom{x}{k-x} = 1$$ For all $x\in\mathbb R_{\ge0}$ and with $\binom{x}{r}=\frac{\Gamma(x+1)}{\Gamma(r+1)\cdot\Gamma(x-r+1)}$ It is obviously ...
4
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148 views

Proving an equation involving binomial coefficients

Prove that $$\sum_{q=0}^v \binom{v}{q}\frac{q!}{v^{q+1}} = \sum_{q=0}^{v-1} \binom{v-1}{q} \frac{(q+2)!}{v^{q+2}}$$ Thanks. Below are what I have tried: Approach 1: $$\sum_{q=0}^{v-1} ...
4
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133 views

Combinatorial Identity

I have to validate the following identity which is defined: $$ \sum_{k=1}^n (-1)^{k-1}*q^{\frac{k(k-1)}{2}} *\frac{\prod_{i=n-k+1}^n(1-q^i)}{\prod_{i=1}^k(1-q^i)} = 1 $$ where $0<q<1$. I ...
4
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119 views

Binomial-like sum involving falling factorials

We know that $\sum_{k=0}^n a^k \frac{n^{\underline k}}{k!} = (1+a)^n$. Is there a known (preferably closed) form for $\sum_{k=0}^n a^k n^{\underline k}$?
4
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71 views

Binomial coefficient sum over top index

I am trying to evaluate a sum over binomial coefficients which is giving me some problems. Specifically I want to calculate: $$\sum_{r=0}^{c-1}\binom{r+n}{n}\frac{1}{c-r}$$ My main thought was to ...
4
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66 views

In Pursuit of a Broader Understanding of Complicated Binomial Coefficient Sums

$$\sum_{k=0}^{n}\binom{n}{k}\frac{k!}{(n+k+1)!}$$ The above identity was posted once before by me, however, all results were obtained numerically exploring the identity rather than understanding ...
4
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157 views

How to transform series of series into series

I need to prove this equation. $$ \sum_{k=0}^{i-2} \left( e \space α(k+1)\space\frac{(-1)^{i+k+2}}{(i-k-2)!} \right) = \sum_{k=0}^{i-2} \frac{(i-k)^k}{k!} \space e^{i-k} (-1)^k\space where,\space α(i) ...
4
votes
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94 views

Binomial transform of a scaled version of the catalan numbers.

I was looking at the mathworld entry for Catalan Numbers http://mathworld.wolfram.com/CatalanNumber.html and was surprised to find formula (11) there: (1) $C_n= \sum_{k=0}^n (-1)^k ...
4
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187 views

Simplifying an expression involving binomial coefficients

Consider $$ \prod_{j=0}^{i-1}\binom{n-2j}{2}\left[1+\sum_{k=0}^{N-1}{\left( \prod_{l=0}^{k} \dfrac{N-l}{M-l}\right)\left(1 + \sum_{m=k+1}^{N-1} \prod_{p=k+1}^m \dfrac{N-p}{M-p}\right)+1}\right] $$ ...
4
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223 views

How fast does $\binom{n}{k}$ grow when $k \le n/2$?

How fast does $\binom{n}{k}$, $n$ fixed, grow when $k \le n/2$? Especially, I'm interested in the growth of the "inverse" of binomial coefficient $B_n(x) := \min \{k:\binom{n}{k} \ge x\}$. EDIT: ...
4
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377 views

p-adic numbers and binomial coefficients

Let $\alpha\in \mathbb{Z}_p$ be an $p$-adic integer and define for $n\in \mathbb{Z}_{\geq 0}$ $${\alpha\choose n} := \frac{\alpha(\alpha-1)\cdot\ldots\cdot(\alpha-n+1)}{n!}.$$ This is again a ...
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70 views

Prime factors of binomials

Is it true that for each $n\geq 2$ there are two primes $p, q$ such that (at least) one of them divides $\binom{n}{k}$ for each $1\leq k\leq n-1$? Examples: For $n=6: \binom{6}{1}=6; ...
3
votes
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90 views

Unusual binomial sum: $\sum_{d=k}^{n} {d \choose k} p^{d}(1-p)^{n-d}$

Does anyone know how to simplify the following sum? It's been giving me and everyone else I've showed it to quite a bit of trouble. I'm quite confident that this should simplify, but I just can't seem ...
3
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46 views

Lucas' Theorem for $p$-adic integers?

Does Lucas' Theorem hold for $p$-adic integers? More specifically, does it specifically hold for the case that, given a $p$-adic integer: $$x = x_0 + x_1p + x_2p^2 + \cdots = \sum_{i=0}^\infty ...
3
votes
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98 views

Generating functions to solve number of integer solution problem

If I have $x_1 + x_2 + x_3 =10$ with $1\leq x_1 \leq 5, \; 2 \leq x_2 \leq 6, \;3 \leq x_3 \leq 9$ I know that I compute $(t^1+\dots + t^5)(t^2 +\dots + t^6)(t^3+\dots +t^9)$ and look at the ...
3
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84 views

What's so special about binomial coefficients that someone decided to organize them in a triangle?

I know that binomial coefficients are related to figurate numbers (which were studied by Greeks a loooong time ago, because of its connections to geometry). I also understand how the Pascal's triangle ...
3
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86 views

Combinatorical interpretation of $\binom{15}{5} = \binom{14}{6}$

I was reading up on Singmaster's conjecture on repeated binomial coefficiencts and I read that $$\binom{15}{5} = \binom{14}{6}$$ Sure, it's possible to prove it non-combinatorically: ...
3
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134 views

Proving $\sum_{k=1}^{n}\binom{n-1}{k-1}{\binom{n+k}{k}}^{-1}=\frac 12$ combinatorially

Question : How can we prove the following equations combinatorially? $$\begin{eqnarray}\sum_{k=1}^{n}\frac{\binom{n-1}{k-1}}{\binom{n+k}{k}}&=&\frac ...
3
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41 views

How to show that $\sum\limits_{k=0}^{\lfloor0.999n\rfloor}\binom{2n}{k} < \binom{2n}{n} $ holds for large n

It seems logical to me since $\binom{2n}{n}$ is in the middle of the row in pascal triangle; therefore, the largest, and for large n the sum adds only the small ones on the left. But I do not have any ...
3
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316 views

How to calculate this triple summation?

I need to calculate the following summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m-j}\choose{k-j}}}{k\choose j}r^{k-j+i}$$ I do not know if it is a well-known summation or not. ...
3
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44 views

Showing that a number is part of sequence A000275 in OEIS

Consider the sequence of integers defined recursively by $c_0 = 1$ and $$ c_p = \sum_{l = 0}^{p-1} (-1)^{p+l+1} \binom{p}{l}^2 c_l $$ for $p \geq 1$. This is sequence A000275 in the online ...
3
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213 views

Proving identity involving sum

I'm stuck trying to prove the following identity: $$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose j}(4S-i-k-3)!(4S-2i-1)}{(4S-i-j-1)!} $$ $$=\frac{(-1)^{j+k}(4S-2k-3)!{k+1 ...
3
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140 views

Prove: $\frac{(2px)!}{((px)!)^2}\equiv\frac{(2x)!}{((x)!)^2}\pmod{p^2}$

How can I prove the following, where $p$ is a prime and $x$ a positive integer? $$\dfrac{(2px)!}{((px)!)^2}\equiv\dfrac{(2x)!}{((x)!)^2}\pmod{p^2}$$ I'm not sure if it is actually true, but I tested ...
3
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231 views

Sums of rows in the Pascal triangle

Assume for simplicity that when $k>n$ we have ${n\choose k}=0$. It is well-known that $\sum_k {n\choose 2k}=\sum_k {n\choose 2k+1}=2^{n-1}$ , i.e. the sum of the odd places in each row in Pascal's ...
3
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178 views

Is this formula for $\zeta(15)$ true?

Apery gave, $\begin{aligned} \zeta(3) &= \frac{5}{2}\,\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k^3\,\binom {2k}k}\end{aligned}$ J. Borwein and D. Bradley found this can be generalized to ...
3
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168 views

proof of formula and calculation sum

Show that following formula is true: $$ \sum_{i=0}^{[n/2]}(-1)^i (n-2i)^n{n \choose i}=2^{n-1}n! $$ Using formula calculate $$ \sum_{i=0}^n(2i-n)^p{p \choose i} $$
3
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92 views

max number of times integer $>1$ appears in pascals triangle

$120, 210 ,3003$ appear $6$ times in Pascal's triangle. $120={10\choose3}={16\choose2}={120\choose1}\\$ $210={10\choose4}={21\choose2}={210\choose1}\\$ ...
3
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334 views

Sylvester's Theorem and Schur Theorem

I'll probably end up asking more programming questions on StackExchange forums than math questions, but I'll lead off with a math question. In my Number Theory class this past semester, I worked on a ...
3
votes
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127 views

Defining a signed involution

We define the displacement of $\pi$ as $\mathrm{disp}(\pi)=\sum_{i=1}^n|\pi(i)-i|$. I know that it's even. Could you help me to find a good signed involution of the set of permutation with ...
3
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322 views

Solving a certain binomial sum involving the floor function

Does anybody know of any techniques to solve the following sum (or a fast way (polynomial in N) to compute it): $$\sum_{i=0}^{N} \binom{N-i}{A} \cdot \binom{\lfloor iP/Q \rfloor + 1}{B}$$ where $P$ ...
2
votes
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39 views

General solution for a combinatorial problem

I want to find a general solution for a problem. I explain the problem with an example. $\underline{Problem}:$We have a matrix $A$ of size $M \times N$, where $M <N$. We choose sub-matrices of ...
2
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33 views

Analyzing a coin tossing game with cheating

Consider a game where you toss $N$ coins, and let $H$ denote the number of heads. Let's say you win the game if $|H - N/2| \geq K$, i.e. if the number of heads deviates east least $K$ from what you ...
2
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33 views

Find $f(n)$ in $\binom {2^n} {n^4} = (f(n)+ o(1))^n$

Task is to find $f(n)$ in the following equation: $\binom {2^n} {n^4} = (f(n)+ o(1))^n$ I've found that the problem is a bit over my head. I'm attaching my partial solution below: With use of the ...
2
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33 views

Evaluate $ \sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ where $n,k$ are fixed

Is there a general way/technique to evaluate $ \sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ in terms of $r$, where we consider $n$ and $k$ fixed natural numbers and $n > k$? (here, ...
2
votes
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41 views

How to prove: $pq=\binom{k}{q}-\binom{p}{q}-\binom{k-p}{1}+1$

For any two distinct primes $p, q$ there is a unique integer $k$ such that: $$pq=\binom{k}{q}-\binom{p}{q}-\binom{k-p}{1}+1$$ Where $k$ is the smallest integer greater than $p$ that is relatively ...
2
votes
0answers
27 views

Proving an inequality having binomial coefficients

Suppose that $0 \leq b \leq b+x < a$. How could I prove the inequality \begin{equation} \left(\frac{a-b-x}{a-x}\right)^x \leq \cfrac{\binom{a-x}{b}}{\binom{a}{b}} \leq \left(\frac{a-b}{a}\right)^x ...
2
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57 views

Sum and binomials

I have this sum ...
2
votes
0answers
99 views

Sum of product of binomial coefficients and exponential function

I would like to know how to obtain (if it exists) a closed form expression of the sum $$S=\sum^{n}_{k=0}2^k{{n+1}\choose k}{{r-n-2}\choose {n-k}}$$ So far, I have tried to use the method of ...
2
votes
0answers
53 views

${n \choose r}=\frac {n!}{r!(n-r)!}$ without using the permutation approach.

I had an idea that would be to first prove Pascal's Rule, $${n \choose r} = {n-1 \choose r-1} + {n-1 \choose r}$$ which can be proved combinatorically whether one particular element(among the $n$) is ...
2
votes
0answers
37 views

Inequality in matroid theory

Working on a proof in matroid theory I found there is a smooth map from an open set of $(\mathbb{C}^{\ast})^{(d−1)(n−d−1)}$ to a disjoint union of tori $(S^{1})^{\binom{n}{d}-n}.$ As a direct ...
2
votes
0answers
55 views

find the the variable that maximizes a function

I have a function that I am trying to find for what input it maximizes. $$ f(n) = {\binom{S}{2}}^{n/S}$$ I need to find the $S$ for which this function maximizes (for infinite $n$). more generally, ...