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Results tagged with combinatorial-identities
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user 46140
1
vote
A $1$-step convolution identity involving the Motzkin triangle
Using different bound variables on the two sides for clarity in the subsequent discussion, the goal is:
$$\sum_{k=0}^{n-1}T(n,k) \, T(n,k+1)=\sum_{j=0}^{n-1}\binom{2n}{2j+1}\binom{2j+1}{j}\frac1{j+2}
…
- 7,226
7
votes
Accepted
Is this combinatorial identity known? (of interest for random matrix theory)
Firstly, exploit the finite support to simplify the limits of the sums. Secondly, split the second sum. We get
$$\begin{align*}A(r,b) =& \sum\limits_{m=1}^{r} (2m-1) {r \choose b-m}{r \choose b+m-1} \ …
- 7,226
1
vote
Curious identity involving the number of perfect matchings of the complete graph
It is somewhat unclear from the question which $=$ indicate known identities and which indicate conjectured identities. I assume from the "it suffices" that what you want is a proof of $$n[n+(n-1)]\cd …
- 7,226
2
votes
Accepted
Could you please confirm or deny two identities involving weighted Stirling numbers of the s...
$R(n, k, -\tfrac k2)$ is just the central factorial number $T(n, k)$. (Given the definition of the central factorial numbers, it may be more natural to use them in your context than $R$).
Consider A13 …
- 7,226