0
votes
1answer
27 views

Value of an iterated sum

I am interested in the number of function evaluations required to numerically evaluate an iterated integral of the form $$ \int_0^t \int_{t_1}^t \cdots \int_{t_{n-1}}^t f(t_1,\ldots,t_n) dt_n\cdots ...
1
vote
2answers
44 views

How to show the identity relating to Matrix

Suppose that $$ A=\begin{bmatrix}a_{11}&a_{21}\\a_{21}&a_{22}\end{bmatrix}, \ \ B=\begin{bmatrix}d&-1\\1&0\end{bmatrix}. $$ and $$A=B^N$$ Show that $$a_{11}=\sum_{i=0}^{[N/2]}(-1)^i ...
3
votes
3answers
109 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 ...
2
votes
2answers
68 views

Why does this sum equal to (4^n -1)

How do I get to this solution? $\sum _{k=1}^n\left(\binom nk 3^{n-k}\right)=\left(4^n-1\right)$ I believe it's connected to this, which I know is true: $\sum \:_{k=1}^n\binom nk=2^n-1$
3
votes
3answers
120 views

How to count the $r$-tuples of subsets of $\{1,\dots,n\}$ that are cyclically disjoint?

I want to count the following, $$\#\{S_1,S_2,\dots, S_r\subseteq[n]\;|\; S_i\cap S_{i+1}=\emptyset \text{ for } 1\leq i\leq r-1 \mbox{ and } S_1\cap S_r=\emptyset\}=A_{n,r},$$ Then $A_{n,1}=2^n$, ...
0
votes
1answer
64 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
72 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
91 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 ...
0
votes
1answer
23 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
55 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.
0
votes
0answers
27 views

Simplifying a combinatorial sum

Show that \begin{align} y\sum\limits_{i=1}^dx^iz^i\sum\limits_{j=1}^iq^{i-j}G_d(x,y,q\mid j) = y\sum\limits_{i=1}^d(x^iz^i+\cdots+x^dz^dq^{d-i})G_d(x,y,q\mid i) \end{align} where \begin{align} ...
1
vote
1answer
33 views

Extracting the coefficient of $x^n$ from a fraction

I need help extracting the coefficient of $x^n$ from a $\frac{1-x}{1-2x}$. So far I have that \begin{align} \frac{1-x}{1-2x} &= \frac{1}{1-2x} - x\frac{1}{1-2x}\\ &= \sum\limits_{k=0}(2x)^k ...
1
vote
1answer
30 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 ...
5
votes
3answers
82 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 ...
1
vote
1answer
31 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
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 $Q=\max\{m,n\}$.
0
votes
1answer
44 views

How to calculate the sum of combinatorial numbers

For my work on an almost completely unrelated field I came across the following formula. I know that I have learned that all in high school. But since this is more than 15 years ago in which I never ...
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
2answers
86 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 ...
0
votes
0answers
59 views

Showing that two sums are equivalent

given \begin{gather} U_d(x,y,q\mid i_1,\ldots,i_k)=\sum\limits_{n,m\geq0}x^ny^m\sum\limits_{\sigma = i_1\ldots i_k\sigma_{k+1}\ldots\sigma_m\in C_{[d]}(n,m)}q^{v(\sigma)}. \end{gather} show ...
0
votes
0answers
40 views

The sum of palindromes from 100 to 900

I'm working with palindromes from $100-999$. I'm having trouble with the step highlighted in red. Can someone explain the algebra to me? Taken from: Discrete and Combinatorial Mathematics: An ...
2
votes
1answer
31 views

What is Vandermonde's formula with multisets?

I need Vandermonde's formula in multi-set form. I modified the original formula but I get a mess with too many letters everywhere, is there a nice representation? Here's the original: $$ ...
2
votes
0answers
61 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 ...
0
votes
1answer
99 views

Big Mathematics Challenge on Set and Summation? [closed]

please be aware that this is not homework. it's past PHD entrance Exam on 2011. Suppose: $$B=\{(A_1,A_2,A_3) \mid \forall i; 1\le i \le 3; A_i \subseteq \{1,\ldots,20\}\}$$ if we have: ...
3
votes
2answers
30 views

Find value of $n$ with given conditions

The 4-digit positive number $n$'s digit sum is $20$. The sum of the first two digits is $11$, the sum of the first and the last digit as well. The first digit is the last digit $+3$. What is the ...
4
votes
2answers
74 views

How to evaluate the sum $\sum_{k = 0}^{n}2^k {{n}\choose {k}}$ [duplicate]

How do I evaluate the sum: $$\sum_{k = 0}^{n}2^k {{n}\choose {k}}$$ I know that $2^k = {n \choose 0} + {n \choose 1} + {n \choose 2} + {n \choose 3}... {n \choose k}$, but I don't know how to proceed ...
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) ...
3
votes
4answers
116 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 ...
0
votes
3answers
35 views

Sum of combination equilvalence

Could someone explain to me why the identity $$ \sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k} $$ holds?
0
votes
0answers
64 views

What's the interpretation of $\sum_{i,j} i \cdot j \cdot \binom{2n}{i}\cdot \binom{2n}{j} \cdot \binom{2n}{3n-i-j}$?

I'm having problems with finding the combinatorial interpretation of this sum: $$\sum_{i,j} i \cdot j \cdot \binom{2n}{i}\cdot \binom{2n}{j} \cdot \binom{2n}{3n-i-j}$$ Can anyone help, please?
0
votes
0answers
42 views

Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
1
vote
2answers
61 views

How to prove this identity [closed]

I would like to prove the following identity without using induction: $$\sum _{ k=1 }^{ n }{ { (-1) }^{ k } {n\choose k} }\cdot k^n=(-1)^n\cdot n!. $$
8
votes
4answers
210 views

Does $\sum_{k=0}^{k=n} {n \choose k} k!$ have a closed form for integers $k,n$?

While doing research in computer system, I came across the following summation: $$S_n = \sum_{k=0}^{n} {n \choose k} k! = \sum_{k=0}^{n} \frac{n!}{(n-k)!}$$ where both $n$ and $k$ are integers. $S_n$ ...
5
votes
1answer
87 views

An inverse binomial summation.

I am looking for a closed form for this summation: $$ \sum_{j=1}^m\frac{r^{-j}}{j{m\choose j}} = \sum_{j=1}^m\frac{r^{-j}}{m{m-1\choose j-1}} = \frac1{rm} \sum_{k=0}^{m-1}\frac{r^{-k}}{{m-1\choose k}} ...
0
votes
0answers
31 views

Simplified formula for the following sum

I have the following sum $$\sum_{n\ge 0}\sum_{0\le \alpha_i\le n+1,\\ \sum_{k=1}^N \alpha_i=n+1} \frac{e^{-(\rho+s)\left(\sum_{k=1}^N \alpha_kT_k\right)}\left(\rho\sum_{k=1}^N ...
3
votes
0answers
117 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 ...
1
vote
0answers
35 views

Simplifying a sum- combinatorics

Hello fellow mathematicians, I have been working avidly as a high school project to prove Legendre's conjecture. The question below and the other questions I have posted are directly linked to a ...
0
votes
2answers
70 views

Evaluate the sum $\sum_{0\leq j < k\leq n}\binom{n}{j}\binom{n}{k}$

Could someone give me a hint on how to do this? I believe I know what the answer to be (I computed some low values and checked on OEIS). However, I was hoping someone would be able to explain to me ...
1
vote
2answers
78 views

Sum of numbers on chessboard.

Consider the squares of an $8 \times 8$ chessboard filled with the numbers $1,2,3,4 \ldots ,64$ in sequential order. If we choose $8$ squares with the property that there is exactly $1$ from each ...
4
votes
7answers
2k views

I have the pattern: 1 + 2 + 3 + 4 + 5 + 6, but I need the formula for it

I'm writing some software that takes a group of users and compares each user with every other user in the group. I need to display the amount of comparisons needed for a countdown type feature. For ...
0
votes
2answers
26 views

Problem finding explicit function without multiple $\sum$s

For $1≤n \in \mathbb{N}$, I want a function $f$ to have the following value: The sum of all possible products $a_1 * a_2 * ... * a_n$, with $a_i \in \{1, 2, 3, 4\}$ and $a_1 ≤ a_2 ≤ ... ≤ a_n$. For ...
0
votes
1answer
58 views

Show $\large\sum\limits_{j=0}^{r}\binom{j+k-1}{k-1}=\binom{r+k}{k}$

Show $\large\sum\limits_{j=0}^{r}\binom{j+k-1}{k-1}=\binom{r+k}{k}$ Hint: Place $r$ balls in $m$ urns, in how many of this arrangements can you find $b$ balls in the first urn. I'm sure that ...
4
votes
4answers
190 views

How to calculate the following sums?

I would like to know of a way to evaluate the following two for arbitrary $n$. $$\sum_{i=1}^ni!\,, \quad \sum_{i=1}^n \frac{n!}{i!}. $$
4
votes
3answers
208 views

How to closed the sum $\displaystyle \sum_{k=0}^n \dfrac{(-1)^k(2k+1)!!}{(n-k)!k!(k+1)!}$

How to closed the sum $\displaystyle S=\sum_{k=0}^n \dfrac{(-1)^k(2k+1)!!}{(n-k)!k!(k+1)!}$ I'm trying divide two cases $n$ odd and $n$ even. I predict that ...
0
votes
0answers
28 views

Is there an upper limit to the number of times a value can occur in a superset?

Given a set of numbers S=(-5,6,9,3,2,-2,), is there an upper limit to the number of times a particular value (say 4) can occur in the sums of all the combinations of these numbers? For example: in ...
5
votes
0answers
82 views

How to prove this indentity $\binom{100}{0}^2-\binom{100}{1}^2+\binom{100}{2}^2-…-\binom{100}{99}^2+\binom{100}{100}^2=\binom{100}{50}$ [duplicate]

I don't know how to prove this identity: $\binom{100}{0}^2-\binom{100}{1}^2+\binom{100}{2}^2-\binom{100}{3}^2+...-\binom{100}{99}^2+\binom{100}{100}^2=\binom{100}{50}$
3
votes
2answers
115 views

Properties of a sequence of sums of binomials

I have encountered the following sequence of alternating sums of binomials and I am wondering whether there is a nicer way to write every element and/or are there some nice properties about it. So, if ...
1
vote
1answer
46 views

Simplifying Sum of Subsets

Given sets $A$ and $R$ such that $R \subseteq A$ and a number $x \leq |A|$, I am trying to simplify the following sum: $$\begin{equation*} \sum_{R \subseteq W \subseteq A : |W| = x} \Big( \sum_{Y ...