combinatorics - Proof of the hockey stick/Zhu Shijie identity $\sum ...
EDIT 01 : This identity is known as the hockey-stick identity because, on Pascal's triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed.
Another Hockey Stick Identity - Mathematics Stack Exchange
2016年5月14日 · Proof of the hockey stick/Zhu Shijie identity $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$ 0 Notation for writing multinomial coefficient as sum of smaller multinomial coefficients
A combinatorial identity - Hockey Stick generalization
2020年1月29日 · There is a well known identity (the so called "Hockey-stick identity") asserting that: $$\sum_{j=0}^m \binom{r+j}{j} = \binom{m+r+1}{r+1}$$ For some proofs see this.
combinatorics - Hockey Stick Identity Summation Proof
2020年10月14日 · For example in the hockey stick identity that OP posted, if j=4, then the first summand would become 0C4, which is undefined. $\endgroup$ – user140161 Commented Oct 17, 2020 at 22:36
Enumerative interpretation of generalized $q$-hockey stick identity
may be used recursively to obtain the hockey stick identity $$ \binom{n+1}{k+1}=\binom{n}{k}+\binom{n-1}{k}+\cdots+\binom{k}{k}. \tag{2} $$ The reason for the name is that if all these binomials are highlighted in Pascal's triangle, they form what looks like a hockey stick. This is a special case of a more general identity,
Proof that $\\sum_{k=0}^n \\binom{k}{p}$ = $\\binom{n+1}{p+1}$
It is more or less famously known as the hockey-stick identity. If you recall Pascal's triangle, notice that each number is the sum of the two above. Now take the orange number $84$ below. It is the sum of the two above it: $84=56+28$. Similarly, $28$ is the sum of the two numbers above it, and then the sum above it etc. so that we end up with
Prove of Vandermonde's identity - Mathematics Stack Exchange
2023年8月2日 · Here we show OPs identity is a Chu-Vandermonde identity in disguise. The following is valid \begin{align*} \color{blue}{\sum_{k=-q+n}^{l-m}\binom{l-k}{m}\binom{q+k}{n}=\binom{l+q+1}{m+n+1}\qquad l,m,n,q\in\mathbb{N}}\tag{1} \end{align*} where we set lower and upper limit of the sum, so that the upper index of the binomial …
Prove $\\sum\\limits_{i=0}^n\\binom{i+k-1}{k …
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Hockey-Stick Theorem for Multinomial Coefficients
Proof of the hockey stick/Zhu Shijie identity $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$ 0 Notation for writing multinomial coefficient as sum of smaller multinomial coefficients
Pascal's Identities for - Mathematics Stack Exchange
2017年9月23日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.