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# Questions tagged [binomial-coefficients]

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### Closed form $\int_{0}^{\frac{r}{2}} {\binom{n}{p} \binom{n-p}{r-2p} 2^{r-2p}}{\binom{2n}{r}^{-1}} \ \text{d}p$

Note: This is exact copy of my Math.SE question, which I am reposting here, as despite bounty it did not receive any answers. Let there be $n$ pairs of shoes in a box. The the probability that from ...
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We all know that $\sum_{k=0}^n\binom{n}k$ is not divisible by $3$. QUESTION. Is it true that the numerator of $a_n$ (in reduced form) is never divisible by $3$? $$a_n=\sum_{k=0}^n\frac1{\binom{n}... 0answers 84 views ### Q-analogue of an inequality Pick integers b\geq a \geq 0 and k\geq j\geq 0. It is not super-difficult to prove the inequality$$ \binom{kb}{ka}^j \geq \binom{jb}{ja}^k. $$This is actually quite a nice inequality that was ... 3answers 190 views ### How to calculate \sum \limits_{k=0}^{m-n} {m-k-1 \choose n-1} {k+n \choose n}? How to calculate$$\sum\limits_{k=0}^{m-n} {m-k-1 \choose n-1} {k+n \choose n}.$$3answers 306 views ### How to calculate: \sum\limits_{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k} How to calculate:$$\sum _{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k}.$$3answers 115 views ### A clean upper bound for the expectation of a function of a binomial random variable I wonder if there is a closed-form, or clean upper bound of this quantity: \mathbb{E}[|X/n-p|], where X\sim B(n,p). 3answers 988 views ### Does the set \{\binom x3+\binom y3+\binom z3:\ x,y,z\in\mathbb Z\} contain all integers? The Gauss-Legendre theorem on sums of three squares states that$$\{x^2+y^2+z^2:\ x,y,z\in\mathbb Z\}=\mathbb N\setminus\{4^k(8m+7):\ k,m\in\mathbb N\},$$where \mathbb N=\{0,1,2,\ldots\}. It is ... 1answer 591 views ### A curious inequality concerning binomial coefficients Has anyone seen an inequality of this form before? It seems to be true (based on extensive testing), but I am not able to prove it. Let a_1,a_2,\ldots,a_k be non-negative integers such that \sum_i ... 0answers 153 views ### A binomial coefficient identity i'm unable to prove the following : \forall n integer \geq 3,  \displaystyle \displaystyle \sum_{s=1}^n \sum_{j=n-s+1}^n \displaystyle \frac{ (\binom n j )^2 \binom {n+j} n }{s-n+j} ( \... 1answer 103 views ### Question about arithmetic binomial coefficient i have a question about the following assertion: let n,j,u  positive integer satisfying  n \geq 5,  1\leq j \leq n-1, \; n+1 \leq u \leq n+j let  d[n]:=\operatorname{lcm}[1,2,..,n] ... 2answers 112 views ### Specific partial sum of even/odd binomial coefficients I have a following sum: S_g=\sum_{k=0}^g k\binom{4g+2}{2k} I can transform it into a different sum S_g=(2g+1)\sum_{k=1}^g\binom{4g+1}{2k-1} What is the closed form or what is the method to ... 1answer 118 views ### Simplify a double summation involving binomial coeficient$$T(N,K)=\sum_{i=2}^{K}\sum_{j=2}^{i}(-1)^{i-j}\binom{i}{j}\frac{j^{N+1}-1}{j-1}$$Is it possible to evaluate the sum for K=10^7 efficiently. If we manage to remove one of the sums, it will be ... 2answers 2k views ### Positive integers written as \binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8 with w,x,y,z\in\{2,3,\ldots\} Let \mathbb N=\{0,1,2,\ldots\}. Recall that the triangular numbers are those natural numbers$$T_x=\frac {x(x+1)}2\quad \text{with}\ x\in\mathbb N. As $T_x=\binom{x+1}2$, Gauss' triangular number ...

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