# Questions tagged [binomial-coefficients]

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### Conjecture on bernoulli numbers and binomial coefficients

Crossposted from https://math.stackexchange.com/questions/4116414/conjecture-on-bernoulli-numbers-and-binomial-coefficients In playing around with some formulas, I have come up with the following ...
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### Sum of binomials with power coefficients

I have a sum of the form $$f(n) = \sum_{i=0}^{ \lfloor \log_2 n \rfloor} \sum_{j=0}^{i} \binom{n}{2^j}.$$ Is there a closed formula or an approximation for this expression? What if $n = 2^m$ is a ...
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### Maximum-average prefix of binomial coefficients

For $k\in\mathbb N$, let $$f(k)=\max_{j\in\{1,\dots,k\}}\left\{\frac{\sum_{i=0}^{j-1}\binom{k}{i}}{j} \right\}.$$ Is there a way to express $f(k)$ explicitly? If not, what is $\lim_{k\to\infty}f(k)$?...
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### Binomial coefficient in a binomial coefficient

I am doing some research in combinatorics, and I found that I have to consider the following binomial coefficient : $$\binom{\binom{i}{j}}{k}$$ (In fact, I have to take the product for fixed $i,k$ ...
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### Is there a combinatorial reason for variable-independence of this binomial-coefficient identity?

Consider the following identity $$\sum_{n=0}^{R-t}\binom{n+\ell}n\binom{R-\ell-n}{R-t-n}=\binom{R+1}{t+1}.\tag1$$ It is relatively easy to give an algebraic or mechanical proof of (1). But, I like to ...
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### An ambitiouser binomial coefficients sum

I asked how to calculate $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$ and got amazing answers. A bit later, however, I figured I needed something rather more complicated: I need to find ...
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### Binomial Coefficients sum [closed]

Any idea on whether or not $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$ has a closed formula on $a$ and $b$ (and on what it is, in case it does)? It is supposed that $b \le a$.
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### Closed form for $\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$

As stated, I wonder if there is a closed form for the generating function $F_{\alpha,\beta}(x):=\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$ where $\alpha,\beta \in\mathbb{N}$. Calling ...
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### What is known about sums of the form $\sum_{n=2}^{\infty}[\zeta(n)-1]^{p}$?

A fair bit is known about rational zeta series. This includes identities like $$\sum_{n=2}^{\infty} [\zeta(n) -1] = 1 .$$ Many more identities can be found in articles by e.g. Borwein and Adamchik &...
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### Two remarkable weighted sums over binary words

This question builds off of the previous MO question Number of collinear ways to fill a grid. Let $A(m,n)$ denote the set of binary words $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_{m+n-2})$ consisting ...
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### Estimating the alternating sum $\sum_{j \ge 1} (-1)^j e^{-j^2} j^k$

I have been trying to get a lower bound on the following alternating sum but without much success: $$\sum_{j=1}^T (-1)^j e^{-j^2} j^k .$$ For small values of $k$, this is easy because the first term ...
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### Are there any identities for alternating binomial sums of the form $\sum_{k=0}^{n} (-1)^{k}k^{p}{n \choose k}$?

In equations (20) - (25) of Mathworld's article on binomial sums, identities are given for sums of the form $$\sum_{k=0}^{n} k^{p}{n \choose k},$$ with $p \in \mathbb{Z}_{\geq 0}$. I wonder whether ...
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### Some conjectural congruences involving Domb numbers

The Domb numbers are given by $$D_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}k\binom{2(n-k)}{n-k}\ \ \ (n=0,1,2,\ldots).$$ Such numbers have combinatorial interpretation, see, e.g., http://oeis.org/A002895....
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### Bijective proof of a combinatorial identity: $\sum\limits_{k=0}^n\binom nk^2 \binom k{n-m}=\binom nm \binom{n+m}m$

Identity \begin{equation} \sum_{k=0}^n\binom nk^2 \binom k{n-m}=\binom nm \binom{n+m}m \tag{1} \end{equation} was used in an answer here. As shown in that answer, (1) easily reduces to \begin{...
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### An inequality related to the numbers of faces of polytopes with d+2 facets

I would like to prove an inequality related to the number of $k$-faces of two $d$-polytopes with $d+2$ facets; see (1) below. Let $r>0$, $s>0$, $t\ge 0$, and $d\ge 2$ be such that $d=r+s+t$. We ...
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### Solve equation involving binomial coefficient

I have a problem that leads to the following equation: $${x \choose k} = N$$ For some unknown $x$ and known constants $k$ and $N$. Here all numbers are natural numbers. I can solve this analytically ...
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### Bounding the probability that two binomials are equal

Note: This question was migrated from this earlier post, where it initially appeared. Following suggestions, I moved this into its own question. Let $B_{n,p}$ denote the usual binomial random ...
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### Alternating binomial sum asymptotics

Let $$S_k=\sum_{j=0}^{\alpha k}(-1)^j\binom{k/2}{j}\binom{\alpha k^2-j k}{k}$$ where $\alpha\in(0,1/2)$ is a constant. I'm interested in understanding the asymptotic behaviour of $S_k$. It would ...
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### Bell polynomial with variables 1 and 0

Let $B_{n,k}(x_1,\cdots,x_{n-k+1})$ be the Bell polynomial. If $x_1=\cdots=x_{n-k+1}=1$, we know that $B_{n,k}(x_1,\cdots,x_{n-k+1})=S(n,k)$, where $S(n,k)$ is the Stirling number of second kind. ...
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### Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$

For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial ...