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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 ...
      1
      vote
      2answers
      111 views

      Monotonicity of $M$-sequence

      Consider the following definition in the second page of this article: For any two integers $k,n\ge 1$, there is a unique way of writing $$n=\binom{a_k}{k}+\binom{a_{k-1}}{k-1}+\dots+\binom{a_i}{i}...
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      264 views

      Voyage into the golden screen (sequence defined by recurrence relation)

      We start from A004718 named "The Danish composer Per N?rg?rd's "infinity sequence", invented in an attempt to unify in a perfect way repetition and variation" $$a(2n) = -a(n), \qquad a(2n+1) = a(n) + ...
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      308 views

      Reciprocal sum of binomials and divisibility by $3$

      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}...
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      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 ...
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      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 $$\sum\limits_{k=0}^{m-n} {m-k-1 \choose n-1} {k+n \choose n}.$$
      3
      votes
      3answers
      306 views

      How to calculate: $\sum\limits_{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k}$

      How to calculate: $$\sum _{k=0}^{n-m} \frac{1}{n-k} {n-m \choose k}.$$
      4
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      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 $X\sim B(n,p)$.
      12
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      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 ...
      8
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      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 ...
      2
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      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} ( \...
      1
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      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]$ ...
      0
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      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 ...
      0
      votes
      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 ...
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      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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