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Results tagged with binomial-coefficients
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user 38620
For questions that explicitly reference the binomial coefficients, Pascal's Triangle, and Binomial identities.
4
votes
0
answers
250
views
How find this binomial-coefficients sum $\sum_{k_{1}+k_{2}+\cdots+k_{d}=n}\binom{n}{k_{1},k_... [duplicate]
Assmue that $d$ is give postive integer numbers,and
$$(x_{1}+x_{2}+\cdots+x_{d})^n=\sum_{k_{1}+\cdots+k_{d}=n} \binom{n}{k_{1},k_{2},\cdots,k_{d}}x^{k_{1}}_{1}x^{k_{2}}_{2}\cdots x^{k_{d}}_{d} …
- 4,280
3
votes
1
answer
462
views
How to prove this combinatorial identity?
If $n \in \mathbb N \setminus \{0\}$ and $x,y,z \in \mathbb R$ such that $x+y+z=n-1$, show that
$$\dfrac{(-4)^n}{\binom{2x}{n}}\sum_{r+s=n,r,s\in Z}\dfrac{\binom{y}{r}\binom{y-a}{r}\binom{z}{s}\binom{ …
- 4,280