Questions tagged [binomial-coefficients]
For questions that explicitly reference the binomial coefficients, Pascal's Triangle, and Binomial identities.
476 questions
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On sums of a prime and a central binomial coefficient
Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that
$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$
...
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Prove by induction a formula involving binomial coefficients
Define $$f(k) = - \frac{\Gamma(\frac{k-1}{2}) \pi^{3/2}}{\Gamma(\frac{k}{2}) 2^{\frac{k-1}{2}}} \sum_{m=0}^{\lfloor\frac{k-3}{4}\rfloor} \binom{\frac{k-1}{2}}{2m+1} \binom{2m}{m} 4^{-m},$$
And let $$M(...
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For which $x, c$ does $\sum_{n=0}^{\infty}\binom{x}{n}\sum_{k=0}^{n} \binom{n}{k}(-1)^{n-k} |k - c|^{\alpha}$ converge?
Consider the Newton series
$$ \sum_{n=0}^{\infty}\binom{x}{n}\sum_{k=0}^{n} \binom{n}{k}(-1)^{n-k}|k - c|^{\alpha} $$
for $x, c\in\mathbb{R}$ and $\alpha\in (0, 1)$. For given values of $c$ and $\...
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References on Power Sums
Consider a recent arXiv preprint 1805.11445. The author of 1805.11445 has done an overview of classical problem of simplifying of power sum
$$\sum_{1\leq k\leq n}k^m, \ (n,m)\geq 0, \ m=\mathrm{const}$...
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An analogue of the sum of binomial coefficients
Let $a$ and $p$ be positive integers, and consider the polynomial $$(1+x+\cdots+x^{p-1})^a = \sum_{i=0}^{a(p-1)} a_ix^i.$$ I'm looking for an asymptotic estimate of $\sum_{i=0}^{b} a_i$. If $p=2$, ...
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Sum of 'the first $k$' binomial coefficients for fixed $N$
I am interested in the function $$f(N,k)=\sum_{i=0}^{k} {N \choose i}$$ for fixed $N$ and $0 \leq k \leq N $. Obviously it equals 1 for $k = 0$ and $2^{N}$ for $k = N$, but are there any other ...
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Proving that a certain factorial double sum collapses to a double-factorial series
While solving a few sums, I came across the following double sum
$$\sum_{k=0}^{n} (-1)^k\frac{(n+k)!}{2^kk!(n-k)!}x^{n-k}\sum_{j=0}^{n+k} \frac{x^j}{j!}$$
which is expected to evaluate to
$$\sum_{k=0}^...
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Closed form solution for a binomial coefficient relation?
In following, $x_{n}$ is a set of given numbers, n = 0, 1, 2, ...,
$y_{n}$ is defined by the following recursive relation of $x_{n}$:
For example:
${\displaystyle {x_{1}=x_{0}y_{1} }}.$
${\...
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Does the (normalized) product of two independent binomial variables converge in distribution to a normal variable?
(I asked this question on MSE 10 days ago, but I got no answer.)
Let $X$ and $Y$ be two independent identically distributed binomial random variables with parameters $n \in \mathbb{N}$ and $p \in (0,1)...
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Proving convexity of the expected logarithm of binomial distribution
I would like to prove that the following function, for an arbitrary integer $n$:
\begin{equation}
\begin{split}
f(x) & =x\cdot E \ \log(1+\text{Binomial(n,x)}) \\
& = x \cdot \sum_{k=0}^{n} \...
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identity involving products of binomial coefficients
I need the following identity to prove something (regarding functions of two variables):
$$\sum_{k=0}^{n-r} (-1)^k \binom{n-k}{k} \binom{n-2k}{n-k-r} = 1$$
Is this well-known? Is there a quick proof?
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Expected value of Taylor series with central moments of binomial variate
I want to understand this entry, but do not understand how the $\mathcal{O}\left(\frac{1}{n^2}\right)$ in the accepted answer comes into play.
I reproduce the question here: We have $x \sim \mathrm{...
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Does the following sum converge?
Does the sum
$$
\lim_{n\to\infty}\sum_{k=0}^{\lfloor\alpha n \rfloor}C_n^k(-1)^k\left(1-\frac{k}{\alpha n}\right)
$$
converge, where $C_n^k$ is the binomial coefficient and $0 <\alpha <1$?
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Fast converging series for $L(2,(\frac{d}{\cdot}))$
Dirichlet's $L$-function plays a central role in analytic number theory. For any integer $d\equiv0,1\pmod4$, let
$$L_d(2):=L\left(2,\left(\frac{d}{\cdot}\right)\right)=\sum_{k=1}^\infty\frac{(\frac dk)...
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Series for $\frac{\log m}{\pi}$ with summands involving harmonic numbers
The classical rational Ramanujan-type series for $1/\pi$ have the following four forms:
\begin{align}\sum_{k=0}^\infty(ak+b)\frac{\binom{2k}k^3}{m^k}&=\frac{c}{\pi},\label{1}\tag{1}
\\\sum_{k=0}^\...
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