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      Questions tagged [binomial-coefficients]

      The tag has no usage guidance, but it has a tag wiki.

      -4
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      0answers
      28 views

      BInomial theorem [on hold]

      Obtain and simplify the first three terms in the expansion of (1- (x/2))^6 in ascending powers of x. Given that the coefficient of x and x^2 in the expression of (2 + ax + bx^2)(1- (x/2))^6 are -5 and ...
      0
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      3answers
      171 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
      281 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
      votes
      3answers
      100 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 $X\sim B(n,p)$.
      11
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      3answers
      932 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
      votes
      1answer
      572 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
      150 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
      vote
      1answer
      98 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
      votes
      2answers
      109 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
      110 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 ...
      14
      votes
      2answers
      1k 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 ...
      6
      votes
      1answer
      282 views

      Closed form expression for a recursion relation with binomial coefficients

      I am interested in the following sequence: $$ T_n = \sum\limits^{n-1}_{k=0} \begin{pmatrix} n \\ k?\end{pmatrix} T_{k}, \?\?\?\ T_0 = C \in \mathbb{N} $$ I would like to express it as a function of n, ...
      1
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      0answers
      76 views

      Has this result about the number of permutations of a given cycle type (or centralizers) been proved?

      I was playing with the cardinality of conjugacy classes of the symmetric groups, which we know is the number of permutations of a given cycle type and there is a natural one-to-one correspondence ...
      2
      votes
      1answer
      155 views

      Number of odd elements in a vanishing sum of binomial coefficients

      Let $n$ be a positive integer, $k$ a non-negative integer and $N(n,k)$ be the number of odd elements among the numbers $\binom{n+k}{j}\binom{-n-k}{n-j}$, $0\le{j}\le{n}$, which sum to $0.$ It seems ...
      1
      vote
      0answers
      66 views

      Shuffling unordered partitions

      Consider the following: Let $\mathcal{A}$ be an unordered partition of $\{1,\dotsc,p\}$, Let $\mathcal{B}$ be an unordered partition of $\{1,\dotsc,q\}$ Let $\mathcal{C}$ be an unordered partition of ...

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