# Tagged Questions

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### A sum involving binomial coefficients and its evaluation using the Gamma function [migrated]

Does anyone know how to prove (or a reference for) the following identity for positive integers $r$: $$\sum_{i=0}^r (-1)^i{r\choose i}\frac{1}{ir+1}= \frac{\Gamma(1+1/r)\Gamma(r+1)}{\Gamma(r+1+1/r)}$$
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### estimating binomial coefficients

There is a beautiful paper on the arXiv by Andrew Suk containing an asymptotic result about the Erdös-Szekeres convex polygon problem. I am struggling with one of the estimates he makes on page 4. He ...
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### Simplify a equation

I have a problem simplifying the summation here: $$\sum_{x=0}^{n}\sum_{y=0}^{x} {n\choose{x}} {x\choose{y}} y!(x-y)!$$ The last three terms can be simplified to x!, so the current summation ...
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### Are there good bounds on binomial coefficients?

Motivated by the central limit theorem, one expects that $$\binom{n}{k} \approx \frac{2^n}{\sqrt{\pi n/2}} \exp\left(-\frac{(k-n/2)^2}{n/2}\right).$$ Computations suggest that the ratio of the two ...
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### Partial sum of binomial coefficients

For some integer $z \ge 2$ and large integer $n$ and $t=\lceil \log n\rceil$, what is an approximate value for the following partial binomial sum? $$\sum_{i=0}^{n-t} \binom{n}{i}z^i .$$ Another ...
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### Can you simplify (or approximate) $\sum_{n=0}^{N-1} \binom{N-1}n \frac{(-1)^n}{n+1} e^{-\frac{n}{2(n+1)}\lambda}$?

Let $\binom x y$ be the binomial coefficient. I am trying to get a better understanding of the sum $$f(N,\lambda)=\sum_{n=0}^{N-1}\binom{N-1}n\frac{(-1)^n}{n+1} e^{-\frac{n}{2(n+1)}\lambda}$$ as a ...
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### Summation of multinominal coefficients with extra bounds on summation indices

My question is related to the sum $$S(n,N) = \sum_{k_1+k_2+...+k_N=n}\frac{n!}{(k_1!)\cdot(k_2!)\cdot...\cdot(k_N!)} = N^n,$$ which is comes from the multinomial theorem:...
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### Asymptotic for binomial sums

Let $S(n, t) = \sum_{k = 0}^n {n \choose k} ^t$. The task is to find asymptotic behavior of $S(n,5)$, $n \to \infty$. Asymptotic for $S(n,0)$ and $S(n,1)$ is very simple. For $S(n,2)$ we can use ...
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### Closed Form Expression for Nested Series Summation?

Just wandering if there are any criteria that can decide whether a finite series summation has closed form or not. for example, In the following nested summation, $n$ is some even integer that will be ...
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### binomial coefficients and irrationals

The following, probably either currently impossible to deal with, or having a negative solution, arose from an ergodic theory question, presumably itself currently intractible. I am not a number ...
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### Prove that expression is integer

Numerical experiments suggest that $\binom{2m}{m + k}\cdot\frac{3m - 1 - 2k^2}{2m - 1}$ is integer for all $-m \le k\le +m$. It means that expression evaluation could be implemented very efficiently, ...
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### binomial/factorial identity mod p

In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result. Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...
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### How to prove this polynomial always has integer values at all integers?

Let $m$ be any positive integer. $$P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}.$$ Question: $P_m(x)$ always ...
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### Closed form for binomial coeff sum

As part of a proof in finite group theory, I'm looking for a closed form for the expression $$\sum_{i=j+1}^{n} \binom{\binom{i}{j}}{2}$$ Any help - especially with reference or proof - would be ...
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### Equation with $q$-binomial coefficients

Let $d\ge2$, and let $q$ be a power of a prime. As usual, define $N(d,q)=\sum_{k=0}^d{d\choose k}_q$. I wonder if there are $d$ and $q$ as above such that $1+N(d,q)=q^{d+1}$. (If the answer is ...
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### Combinatorial identity and Fuss-Catalan numbers

I would like to show that $$\lim_{N\to\infty}\frac{1}{N^{np+1}}\frac1{p!}\sum_{j=0}^{p-1}(-1)^j\binom{p-1}{j} \left(\frac{\Gamma(N+p-j)}{\Gamma(N-j)}\right)^{n+1} =\frac1{np+1}\binom{(n+1)p}{p},$$ ...
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### Asymptotic of a certain double sum involving binomial coefficients

Consider sums of the form $S(n)=\sum^{n}_{m=0}\sum^{m}_{k=1}2^{2k+m+1}{n-m+k+1 \choose 2k+2}{m \choose k}$ I am interested in the asymptotics of $S(n)$ as $n\to \infty$. More precisely I would ...
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### An inequality concerning non-negative integer matrices with constant row and column sums

[I posted this question on math.stackexchange a few weeks back, but no luck there so far: http://math.stackexchange.com/questions/1095659/an-inequality-concerning-non-negative-integer-matrices-with-...
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### Quest for a human proof of a $q-$binomial identity

Let $$f(n,k) = \sum\limits_{j = - k}^k {{{( - 1)}^{k - j}}} \binom{n-j}{k-j}\binom{n+j}{k+j}.$$ Then $f(n,k)=\binom{n}{k}$ because it satisfies $f(n,k)=f(n-1,k)+f(n-1,k-1)$ and the obvious ...
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### What is $\sum_{i=0}^{n}\binom{n}{i}^3$?

We know that $$\sum_{i=0}^{n}\binom{n}{i}=2^n$$ and that $$\sum_{i=0}^{n}\binom{n}{i}^2= \binom{2n}{n}$$ what about $$\sum_{i=0}^{n}\binom{n}{i}^3$$ ?
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### Simplest form for sum of Binomial Expressions

How difficult is the problem of reducing the number of terms in a sum of binomial expressions? Formally: Given $a_1, a_2, a_3, … a_n$, and $b_1, b_2, b_3, ... , b_n$, where $a_i, b_i \in \mathbb{Z}$, ...
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### Upper bound of sum of binomial coefficients

I am looking for an upper bound - up to constant factor - for: $\sum_{k=t}^{t+l} {n \choose k} \cdot 2^{-n}$ where: The values of $t$ are between: $\frac{n}2+\sqrt{n} \leq t \leq \frac{9n}{10}$. (...
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### How find this binomial-coefficients sum $\sum_{k_{1}+k_{2}+\cdots+k_{d}=n}\binom{n}{k_{1},k_{2},\cdots,k_{d}}^2$ [duplicate]

Assmue that $d$ is give postive integer numbers,and $$(x_{1}+x_{2}+\cdots+x_{d})^n=\sum_{k_{1}+\cdots+k_{d}=n} \binom{n}{k_{1},k_{2},\cdots,k_{d}}x^{k_{1}}_{1}x^{k_{2}}_{2}\cdots x^{k_{d}}_{d}$$ ...
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### Does $\binom{2n}{n} \equiv 2 \pmod p$ ever hold?

Well, the title does not tell the whole story; the complete question is: Are there any primes of the form $p=2n(n-1)+1$, with integer $n\ge 1$, such that $$\binom{2n}{n} \equiv 2\pmod p ?$$ ...
I have a question regarding the zeros of the following polynomial, based on partial rows of pascal's triangle, $$f(x)=\sum_{k=a}^n\binom{n}{k}(-1)^{k}x^k,$$ where $a,n\in \mathbb{Z}^+,n>a.$ ...