# 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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### 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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### Zero sum of binomial coefficients

Is there a function $p:\mathbb N\to \{ 1,-1 \}$ and a fixed $N\in \mathbb N$ such that for every $n \geq N$ we get: $\sum _{i=0} ^{n} p(i)\binom {n}{i}=0$ ? Obviously $p(i)=(-1)^i$ works for $N=1$...
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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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### Bounded convolutions with binomial coefficients

I need to figure out a nice family of decaying functions such that $\sum_{d=2}^k {k \choose d} f_k(d) \leq 1/k$ and $f_k(d)\geq f_k(d+1)$ How can I figure out what good candidates could be? Any ...
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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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### 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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### 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 ...