# Questions tagged [combinatorial-identities]

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### Bilinear recurrence relation between even Bernoulli numbers

Throughout this question $n$ is a positive integer greater than 1. Consider the following well-known identity by Euler, $$\sum_{k=1}^{n-1} \binom{2n}{2k}B_{2k}B_{2n-2k}=-(2n+1)B_{2n}.$$ Rather ...
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### “Non-associative” standard polynomials

I saw somewhere (I appreciate if anyone has any references to proof of this fact) that if $A$ is a finite dimensional associative algebra such that $\textrm{dim}(A)<n$, then $A$ satisfies the ...
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### Showing this formula counts these things

I'm writing an article, and I got stuck trying to prove that some numbers are positive. I have a relatively good intuition for guessing what an expression is counting, but in this case I'm not being ...
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### Identity with binomial coefficients and k^k

In process of doing some computations on Hilbert schemes, I stumbled across the following identity, for $k \ge 2$: $$k^{k-3} = \frac{1}{2} \sum_{i=1}^{k-1} \binom{k-2}{i-1} i^{i-2} (k-i)^{k-i-2}$$ ...
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### One trig “survives” a binomial summation: why?

I've seen many trigonometric identities but here is one that I encountered for which I did not find a reference. In case you wonder where this came from, I was investigating certain $q$-series in ...
I stumbled on the following identity, which has been checked numerically. Question. Is this true? If so, any proof? $$\sum_{j=0}^{\lfloor\frac{k}2\rfloor}\binom{n-2k+j}{j,k-2j,n-3k+2j} =\sum_{j=0}... 2answers 282 views ### Identity with Pochhammer and harmonic numbers This came out of some work on the digamma function. Let (x)_k=x(x+1)\cdots(x+k-1) denote the Pochhammer symbol. Then, Question. Can you prove/disprove this identity?$$\pmb{\frac{(\frac12)_j^...
The sequence $a_n=\sum_{k=0}^n\binom{n}k^24^k$ is listed on OEIS along with a couple of combinatorial interpretations. What interested me at the moment is the plethora of binomial single-sums for the ...