# Questions tagged [binomial-coefficients]

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### Could you please confirm or deny two identities involving weighted Stirling numbers of the second kind?

In the paper  below, among other things, Carlitz introduced weighted Stirling numbers of the second kind $R(n,k,r)$. He also proved that the numbers $R(n,k,r)$ can be generated by \begin{equation*}%...
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### Evaluations of two new series involving Lucas $v$-sequences

Let $A$ and $B$ be integers. The Lucas $v$-sequence $v_n(A,B)\ (n=0,1,2,\ldots)$ is defined by $v_0(A,B)=2,\ v_1(A,B)=A$, and $$v_{n+1}(A,B)=Av_n(A,B)-Bv_{n-1}(A,B)\ \ \ (n=1,2,3,\ldots).$$ From the ...
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### Function maximized by $\left\{\left\lfloor\frac np\right\rfloor,\dots,\left\lfloor\frac{n+p-j}p\right\rfloor\right\}$

Since this MSE question didn't find any suitable answers, I decided to post it here. I was trying to maximize the function $$f(r)=\binom nr\cdot 2^{n-r}$$ This can be done by the standard technique of ...
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### When do binomial coefficients sum to a power of 2?

Define the function $$S(N, n) = \sum_{k=0}^n \binom{N}{k}.$$ For what values of $N$ and $n$ does this function equal a power of 2? There are three classes of solutions: $n = 0$ or $n = N$, $N$ is odd ...
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### Closed form for a binomial product sum

Is there any closed formula for the binomial product sum \begin{align*} \sum\limits_{\substack{i_1> i_2> \cdots > i_k\\i_1, i_2, \cdots, i_k \in \{n-j+1, n-j+2, \cdots, n-1\}}}\binom{n}{i_1}\...
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### Prove the identity $2(n-1)n^{n-2} = \sum^{n-1}_{i=1}\binom nii^{i-1}(n-i)^{n-i-1}$ [closed]

The given identity: $$2(n-1)n^{n-2} = \sum^{n-1}_{i=1}\binom nii^{i-1}(n-i)^{n-i-1}$$ It seems to be a binomial coefficient problem, but I have tried many ways. There are no more ideas how to prove it....
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### Chebyshev polynomials and ballot numbers

I have asked this question a short time ago on mathstackexchange, but it has already fallen into the abyss of answered and uncommented questions. So I take the risk to ask it on mathoverflow. Playing ...
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### A p adic limit of a binomial coefficient

Let $0 \leq a \leq p^n$ be a number coprime to p. Consider the following sequence of binomial coefficients: $$B_k = \binom{p^{n+k}}{p^ka}$$ as $k\to \infty$. If I did the computation right, the p-...
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