1
vote
1answer
40 views

Binomial Congruence (mod 5) Identity

I've got a (hard?) Putnam-style problem that I've been given to look at . . . I've never worked any problem even vaguely like this, but my director thinks I should be able to do it. I doubt it (100% ...
0
votes
1answer
44 views

How to evaluate $\sum_{k=0}^{n} \alpha^k \binom{n}{k}$?

I am trying to show that the function that satisfies $f^\prime(x)=f(x)$ with $f(0)=1$ behaves in an exponential way (in other words, I want to justify writing it as $e^x$). I need to show that: $$ ...
2
votes
1answer
25 views

Sum of combinations of the n by consecutive k

In a book, I found that the sum of combinations: $\binom{n}{k} + \binom{n}{k+1} +\cdots+ \binom{n}{n}$, where k starts from 0, equals $2^n$. It is possible to express this statement via sum: $2 + ...
2
votes
2answers
55 views

Alternating sum of binomial coefficients is equal to zero [duplicate]

Prove without using induction that the following formula:$$\sum_{k=0}^n (-1)^k\binom{n}{k}=0$$ is valid for every $n\ge1$. Progress For each odd $n$ we can use the ...
3
votes
0answers
75 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 ...
-1
votes
1answer
31 views

Binomial Coefficients (2,1) [closed]

What are binomial coefficients? Can someone explain. For example: (2,1). or what (2n,n) means
2
votes
2answers
73 views

Product rule for simplex numbers

The $n$th triangular number is defined as $T_2(n) = n(n+1)/2$, and there is an interesting product rule for triangular numbers: $$T_2(mn) = T_2(m)\,T_2(n) + T_2(m-1)\,T_2(n-1).$$ The tetrahedral ...
3
votes
3answers
122 views

Evaluation of a sum of $(-1)^{k} {n \choose k} {2n-2k \choose n+1}$

I have some question about the paper of which name is Spanning trees: Let me count the ways. The question concerns about $\sum_{k=0}^{\lfloor\frac{n-1}{2} \rfloor} (-1)^{k} {n \choose k} {2n-2k ...
3
votes
3answers
57 views

difficult problem about binomial coefficients

If $r,m,n\in \mathbb N$ so that $r\le \min \{n,m\}$, then $$\binom{n+m}{r} = \binom{n}{0}.\binom{m}{r}+\binom{n}{1}.\binom{m}{r-1}+...+\binom{n}{r}.\binom{m}{0}.$$ If $\min \{n,m\} < r$, then how ...
0
votes
0answers
20 views

Bounding a specific function of binomial coefficients

While trying to directly prove the existence of expander graphs (e.g. http://www.cs.toronto.edu/~avner/teaching/S6-2414/TUT2.pdf), one uses the following inequality: $$\sum_{s=1}^{n/2} ...
4
votes
3answers
156 views

Closed form of a sum of binomial coefficients?

I have the following function: $T_n(d)=\sum\limits_{k=\frac{n-d}{2}}^{\lceil \frac{n}{2} \rceil}{k\choose \frac{n-d}{2}}$ ${n \choose 2k}$, where $n,d\in \mathbb{N}^0$, and $n,d$ have the same ...
4
votes
4answers
70 views

Conjecture for product of binomial coefficient

Is it true that for any $n, k\in\mathbb N$ $$\frac{(kn)!}{k!(n!)^k} = \prod_{l=1}^k {{ln-1}\choose{n-1}} \quad?$$ I tested it for some small $k$ and $n$, but I don't know how to prove that it is true ...
0
votes
3answers
101 views

Given the sequence $3, 4, 11, 16, 42\ldots $ how can I derive a general formula for it?

Given a sequence $3, 4, 11, 16, 42\ldots $ how can I derive a general formula for this sequence? Is there any optimised approach? My approach: the given series is equal to summation of $\binom{n}{k}$ ...
6
votes
4answers
189 views

Truncated alternating binomial sum

It is easily checked that $ \sum_{i = 0}^n (-1)^i \binom{n}{i} = 0$, for example by appealing to the binomial theorem. I'm trying to figure out what happens with the truncated sum $\sum_{i=0}^{D} ...
0
votes
1answer
65 views

Multiple sum involving binomial factors

Let $n$ and $m$ be positive integers and let $0 \le j \le n-m-1$. Show that: \begin{align} \sum\limits_{l=m}^{n-j-1} \binom{n-l-1}{j} \binom{l}{m} \binom{n+l}{j} &=\sum\limits_{p=0}^j ...
2
votes
1answer
75 views

A double sum with combinatorial factors

Let $n$, $p$ and $j$ be integers. As a byproduct of some other calculations I have discovered the following identity: \begin{equation} \sum\limits_{p=0}^{j} \sum\limits_{p_1=0}^j \binom{p+p_1}{p_1} ...
3
votes
1answer
37 views

How to simplify a sum with binomial coefficients multiplied by $k^3/2^k$?

The sum is $$\sum_{k=5}^{\infty}\binom{k-1}{k-5}\frac{k^3}{2^{k}} $$ The first thing I thought of was the binomial coefficient. So I re-indexed it ...
1
vote
3answers
92 views

Simplifying $\displaystyle\sum_{k=0}^{20}(k+4)\binom{23-k}{3}$

In trying to simplify my answer to a problem posted recently, I am trying to show that $\displaystyle\sum_{k=0}^{20}(k+4)\binom{23-k}{3}=8\binom{24}{4}$. I know that ...
2
votes
1answer
52 views

Simplifying a generating function in two variables with two binomial coefficients

I'm trying to to make the below expression simpler, and it would be great if it could be expressed as something like $(x+y)^k$. $$ \sum_{i=0}^k\binom{n+1}i\binom{m+1}{k-i}x^iy^{k-i} $$ The number ...
1
vote
2answers
45 views

Upper bound on $ \binom{a}{m+1}\sum ^m_{j=0} \binom{a-m-1}{j}/\binom{b}{j+m+1}$

Given $a,b,m$ such that $0<2m<a<b$. I would like to find out upper bound of $$S = \binom{a}{m+1}\sum ^m_{j=0} \frac{\binom{a-m-1}{j}}{\binom{b}{j+m+1}}$$ Anyone can help me please? Thank you ...
0
votes
1answer
25 views

Probability $\sum_{j=n+1}^{2n+1} {M \choose m+1}{M-m-1 \choose j-m-1}/{N \choose j} $

I have a prob. problem: A school has $N$ students in which $M$ students are leader (of each class in school), and $N>M$. There are $2n+1$ balls in the black box including $n+1$ blue balls and $n$ ...
1
vote
1answer
56 views

An upper bound and simplification for expression

I would like to find the upper bound (or simplification) of this expression: $$\sum_{j=1}^{n+1}\sum_{i=0}^{j-1} a^{j+i} {j+i \choose i}{n+1\choose j}{n \choose i}/{2n+1 \choose j+i}$$ where ...
0
votes
2answers
63 views

Simplify the expression of binom

Any one knows how to simplify this expression or finding upper bound of this expression: $$\sum_{j=1}^{n+1}\sum_{i=0}^{j-1} a^{j+i} {j+i \choose i}$$ where $0<a<1$ is constant. Thanks a lot.
1
vote
1answer
31 views

Upper bound of $\sum\limits_{j=1}^{n+1} \sum\limits_{i=0}^{j-1}{n+1 \choose j}{n \choose i}$

I would like to find max (or sup.) of the sum: $$S=\sum\limits_{j=1}^{n+1} \sum\limits_{i=0}^{j-1}{n+1 \choose j}{n \choose i}.$$ I found $S\le \frac{1}{\sqrt{\pi n}}.2(n+1).4^n$ but It seems it's ...
1
vote
1answer
41 views

Is there a set of 69 length-6-sets out of 46 numbers [1..46] so that those length-6-sets “cover” all possible 1035 length-2-sets of 46 numbers?

1.) For this question, we have 46 numbers (balls, cards, whatever): {1,2,3,4 .... 45,46} ======================= 2.) Each length-6-set of 46 numbers ( e.g. {1,2,3,4,5,6} or {1,13,16,17,32,46 } ...
5
votes
3answers
86 views

Ordered partitions of an integer (with a twist)

I would like to know how to prove (preferably algebraically) that $P_1(2,n)=F_{2n+1}$, where $P_1(2,n)$ is what I define to be the number of ordered partitions of an integer, where the number $1$ has ...
3
votes
1answer
44 views

How to calculate the number of integer solution of a linear equation with constraints?

If an equation is given like this , $$x_1+x_2+...x_i+...x_n = S$$ and for each $x_i$ a constraint $$0\le x_i \le L_i$$ How do we calculate the number of Integer solutions to this problem?
3
votes
1answer
72 views

Upper bound of $S=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$

EDIT: How can I find a good upper bound to this quantity ? $$S_{n,m}=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$$ where $P=\min\{m,n\}$ et $Q=\max\{m,n\}$.
7
votes
6answers
166 views

Finding $\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $

Help me to simplify:$$\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $$ I got a hunch that it will depend on whether $n$ is a multiple of $6$ and equals to $\frac{2^n+2}{3}$ when $n$ is a ...
3
votes
1answer
48 views

Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$ \begin{array}{cccccccccc} 1 & = & ...
3
votes
2answers
88 views

Prove by induction that $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ is decreasing

I want to prove that the following sequence is monotonously decreasing: $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ I think it should be ...
5
votes
1answer
118 views

Combinations mod $n$ property

So after some "fooling around" I came across this property in Pascal's triangle (which seems to repeat, and makes a lot of sense): $\begin{pmatrix} n \\ k \end{pmatrix} \mod n = \begin{cases} n ...
1
vote
0answers
38 views

A Vandermonde like identity for binomial coefficients

The Vandermonde identity is given by $ \left(\begin{matrix} m + n \\ j \end{matrix}\right) = \displaystyle\sum_{j=0}^k \left(\begin{matrix} m \\ j \end{matrix}\right)\left(\begin{matrix} n \\ k-j ...
1
vote
1answer
137 views

What is the count of the strict partitions of n in k parts not exceeding m?

Lets say we had a $k,m,n \in \mathbb{N}$ where $k < m \le n$. How many different sets $X_1,..,X_m$ with $|X_i|=k$ for $i=1,..,m$, where the sets do not include duplicates, for which the sum of ...
2
votes
0answers
64 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 ...
3
votes
0answers
75 views

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

I was reading up on Sigmaster'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: ...
2
votes
2answers
62 views

Alternating sum of a simple product of binomial coefficients

I would like to evaluate the following alternating sum of products of binomial coefficients: $$\sum_{k=0}^{m} (-1)^k \binom m k \binom n k .$$ I had the idea to use Pascal recursion to re-express ...
0
votes
2answers
70 views

Show that $\binom{n}{k}< \binom{n}{k+1}$ if and only if $k < (n-1)/2$ [closed]

Show that $\binom{n}{k} < \binom{n}{k+1}$ if and only if $k < \frac{n-1}{2}$ and then use this to deduce that the maximum of $\binom{n}{k}$ for $k=0,1,\dots,n$ is $\binom{n}{\lfloor ...
1
vote
1answer
20 views

Proving an identity involving binomial coefficients and fractions

I've been trying to prove the following formula (for $n > 1$ natural, $a, b$ non-zero reals), but I don't know where to start. $$\sum_{j=1}^n \binom{n-1}{j-1} \left( \frac{a-j+1}{b-n+1} \right) ...
2
votes
1answer
26 views

Binomial thereom to figure out coefficents

Use the binomial theorem to find the coefficient of $x^8y^5$ in $(x + y)^{15}$ My textbook shows how to do this looking at the coefficents of Pascal's triangle but, I know theres another way using ...
5
votes
1answer
89 views

Binomial Congruence

How can we show that $\dbinom{pm}{pn}\equiv\dbinom{m}{n}\pmod {p^3}$ for positive integers m and n and p a prime greater than 5? I can do it for mod p^2 but Im stuck here.
1
vote
0answers
49 views

Is there a sharper bound than exponential for $\sum_{k\ge0}\frac{m!(k+n-m)!}{(k+n)!}\frac{s^k}{k!}$?

I am trying find a bound for an expression and I am getting something not quite as convenient as I hoped. Going through my calculations again I think that the only place I use a non sharp bound is ...
1
vote
0answers
23 views

Equation for those level curves?

Related to this question Let $n\in\mathbb{N}^*,\alpha\in\mathbb{R}$. What would be the equation $y=f(x)$ for the curve defined by $\ln\binom{N-y}{x}=\alpha$ That's how they look : TL;DR : What ...
2
votes
0answers
45 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
1answer
79 views

A generalized combinatorial identity for a sum of products of binomial coefficients

I have the following question. For given natural numbers $n$ and $d$, let $a_1,a_2,..., a_r$ be fixed integers such that $a_1+\cdots+a_r=d$. Let $A=\{(i_1,..,i_r)~|~0\le i_j\le n~ \text{and}~ ...
0
votes
0answers
41 views

A sum of powers of binomials

For $n$ and $k$ non-negative integers, let $$F(n,k) = \sum_{i=0}^{n}\binom{n}{i}^k.$$ For example, $F(n,0)=n+1$, $F(n,1)=2^n$ and $F(n,2)=\binom{2n}{n}$. Does there exist a general formula for ...
3
votes
4answers
118 views

Proving Combinatorical Summation: $n!=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^n$ [duplicate]

been stuck with this question for the last few hours, any help would be appreciated. $$ {\large n! = \sum_{k = 0}^{n}\left(-1\right)^{k}{\,n\, \choose \,k\,} \left(\,n - k\,\right)^{n}} $$ what I ...
2
votes
0answers
32 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 ...
1
vote
0answers
30 views

Number of draws that contain at least $k$ red coins

Assume there are $n$ coins in an urn from which $r$ are read. What is the number of draws of $r$ coins that contain at least $k$ red coins? It is obvious that there are ...
0
votes
2answers
46 views

Coefficient of polynomials

Could someone explain to me why $$ [x^{24}](1-2x^6)^{-31} = 2^4 \binom{4 + 31 - 1}{31 - 1} \, ? $$ Reads: The coefficient of $x^{24}$ in $(1-2x^6)^{-31} =$ ...