All Questions
Tagged with binomial-coefficients asymptotics
37 questions
13
votes
5
answers
1k
views
Asymptotics of a Bernoulli-number-like function
Tony Lezard asked me the following question which seemed like it should not be too hard but which I did not immediately see how to answer. Define $f(n,k)$ recursively by $f(1,k) = 1$ and
$$f(n,k) = \...
- 82.7k
12
votes
4
answers
6k
views
Estimating a partial sum of weighted binomial coefficients
There is a well-known estimate for the sum of all binomial coefficients $\binom{n}{k}$ satisfying $k \leq \alpha n$ for some $\alpha$ satisfying $0 < \alpha \leq 1/2$:
$$ \sum_{k=0}^{\alpha n}\...
- 491
9
votes
3
answers
2k
views
Asymptotics of multinomial coefficients
Binomial coefficients have a well known asymptotics (https://en.wikipedia.org/wiki/Binomial_coefficient#Bounds_and_asymptotic_formulas) given by $$\binom nk\sim\binom{n}{\frac{n}{2}} e^{-d^2/(2n)} \...
- 1,826
9
votes
2
answers
791
views
Asymptotic difference between a function and its "binomial average"
(I posted this question on Math.SE a few weeks ago. I got a few comments, but nothing definite, and so I thought I would try MO.)
The origin of this question is the identity
$$\sum_{k=0}^n \binom{n}{...
- 3,283
9
votes
1
answer
1k
views
Partial sums of the Chu--Vandermonde identity
I am interested in finding a lower bound of the sum:
$$\sum_{i=0}^d \left(\genfrac{}{}{0pt}{}{n}{i}\right)
\left(\genfrac{}{}{0pt}{}{m}{k-i}\right)$$
when $d < k$ (and assuming both $n\geq k$, $m\...
- 91
6
votes
3
answers
241
views
Asymptotic expression for $j$ which satisfies $\binom{n}{j}/j! \sim k$ as $n\to\infty$
Suppose $k>0$ is some fixed constant, and $n$ is a positive integer tending to infinity. Find $j\equiv j(n,k)$ such that
$$ \frac{\binom{n}{j}}{j!} \sim k. $$
The asymptotic expression for $(n!)^...
- 386
6
votes
2
answers
912
views
Closed form or/and asymptotics of a hypergeometric sum
Dear mathematicians,
I am a computer scientist wandering in the deep sea of combinatorics and asymptotics to pursue a recent interest in average case analysis of algorithms. In doing so, I designed ...
6
votes
1
answer
290
views
What is the growth rate of the products of binomial coefficients?
Question 1: Are the following empirically observed relationships true
$$
{n \choose 1^a}{n \choose 2^a}{n \choose 3^a}\cdots {n \choose m^a}
\sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{a+3}\bigg)
$$
...
- 5,470
5
votes
2
answers
550
views
Can you simplify (or approximate) $\sum_{n=0}^{N-1} \binom{N-1}n \frac{(-1)^n}{n+1} e^{-\frac{n}{2(n+1)}\lambda}$?
Let $\binom x y$ be the binomial coefficient. I am trying to get a better understanding of the sum
$$
f(N,\lambda)=\sum_{n=0}^{N-1}\binom{N-1}n\frac{(-1)^n}{n+1} e^{-\frac{n}{2(n+1)}\lambda}
$$
as a ...
- 345
5
votes
2
answers
379
views
Asymptotic rate for $\sum\binom{n}k^{-1}$
This MO question prompted me to ask:
What is the second order asymptotic growth/decay rate for the sum
$$\sum_{k=0}^n\frac1{\binom{n}k}$$
as $n\rightarrow\infty$?
- 43.2k
5
votes
1
answer
685
views
Summing ratio of ratio of partial sums of binomial coefficients
I would like to approximate the following when $n \gg k$.
$\sum_{y = k + 1}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m} (y - 1)}{\sum_{m = 0}^k {y - 1 \choose m}}.$
The formula can be re-written ...
- 83
5
votes
1
answer
295
views
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 ...
- 674
4
votes
1
answer
695
views
Approximation a sum involving log and binomial coefficient
I am wondering about the asymptotic approximation of the following expression:
$$S=\sum^{N}_{i=0}\log\Bigg[\binom{\binom{N+1}{i}}{t_i}\Bigg]$$
where
$$t_i=\binom{N}{i}-\binom{N-k}{i-k}+\binom{N-k}{i-...
- 155
3
votes
1
answer
222
views
Asymptotic for binomial sums
Let $S(n, t) = \sum_{k = 0}^n {n \choose k} ^t$.
The task is to find asymptotic behavior of $S(n,5)$, $n \to \infty$.
Asymptotic for $S(n,0)$ and $S(n,1)$ is very simple.
For $S(n,2)$ we can use ...
- 33
3
votes
3
answers
542
views
Estimating a sum involving binomial coefficients [refined]
Having some work done, here is a refined version of my initial question.
For integer $m>0$ and $0\le q\le m$, consider the sum
$$ S(m,q) = \sum_{i=0}^{m-q} \binom{m}{i} \binom{m-i}{q}^2. $$
I ...
- 550
3
votes
1
answer
664
views
(Asymptotics of) Sum involving alternating sign Chu-Vandermonde
While considering eigenvalues of a certain Cayley graph, I came across the following sum:
$$\sum_{r=0}^{d}\sum_{i=0}^{r} (-1)^{i} \binom{w}{i}\binom{n-w}{r-i}$$
where $d$, $w$, and $n$, are all ...
- 618
3
votes
2
answers
2k
views
alternating sum of binomial coefficients
I would like to know a closed formula for
$\sum_{j=0}^{p-n } (-1)^j\binom{n^2}{p-n-j}
\binom{n+j-1}j\binom{2n+j}{n+j+1}$, especially in the
case $p$ is near $n^2/2$. Similarly, I would like a closed ...
- 835
3
votes
0
answers
585
views
Sum of binomial coefficients weighted by a lower incomplete regularized gamma function
The following summation turned up in the course of my research:
$$S_n=\sum_{k=0}^n {n \choose k}\lambda^k P(k,t)$$
where $P(k,t)=\frac{1}{\Gamma(k)}\int_{0}^t e^{-x}x^{k-1}dx$ is the lower ...
- 907
3
votes
0
answers
401
views
Weighted sum of ratio of partial sum of binomial coefficients
I would like to approximate the following sum when $n \rightarrow \infty$ and $n \gg k$,
$\sum_{x = k}^n \sum_{y > x}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^k {y - 1 \choose ...
- 83
2
votes
1
answer
554
views
Asymptotic of a sum involving binomial coefficients
Good evening, I'm trying to find an asymptotic of this sum:
$$\sum_{j=0}^n (-1)^j {n \choose j} (n - j)^n = n^n - {n \choose 1} (n - 1)^n + {n \choose 2} (n - 2)^n + ... + (-1)^n {n \choose n} (n - ...
- 41
2
votes
2
answers
566
views
Asymptotic behaviour of sequence
I am interested in the sequence
$$a(n)=\sum_{k=0}^n {p(n-k) \choose k}$$
where $p(n)$ is a polynomial equation.
When $p(n)=n$ this reduces to the Fibonacci sequence, but what about when $p(n)$ is ...
- 53
2
votes
1
answer
235
views
Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution
$\mu=1+\epsilon$ where $\epsilon>0$ holds.
1.Is there a good bound for $$T=\frac{\sum_{i=-\sqrt{\mu n\ln n}}^{\sqrt{\mu n\ln n}}\binom{n}{\frac n2 +i}^2}{2^n}?$$
This quantity can be ...
- 1,826
2
votes
1
answer
147
views
Tight sublinear estimates for a triple partial binomial summation
Is there tight estimates for the following logarithmic summation ($\gamma,\gamma'\in(0,1)$ and $\mu,\mu'>0$)
$$\log_2\Bigg(\sum_{t=\frac{n^{}}2-n^\gamma\sqrt{\mu\ln n}}^{\frac{n^{}}2+n^\gamma\sqrt{...
- 1,826
2
votes
1
answer
580
views
Asymptotic upper bound for partial binomial-like sum
I want to upper bound the quantity $$\sum_{i\le \alpha n} \binom{n}{i}\lambda^i$$, where ${\lambda>1}$, $0<\alpha<1$. It is not the same as partial sum of binomial coefficients. An asymptotic ...
- 121
1
vote
2
answers
165
views
How fast does this summation grow?
$n,i\in\mathbb N$.
The summation in question is
$$\sum_{k=1}^n\prod_{l=1}^k\binom{2^n}{2^l}^i.$$
How fast does this grow? I am specifically looking at $i=1,2$.
- 13.9k
1
vote
1
answer
142
views
Asymptotics on sum of product of binomial coefficients
I'm interested in the behavior of the summation
$$S(a,b)=\sum_{k\ge 0}\binom{a-k}{k}\binom{b}{k}.$$
For a fixed $\delta>0$, I would like asymptotic bounds on $S(a,\delta a)$.
With $\delta=1$, this ...
- 541
1
vote
1
answer
296
views
$\lim_{k\to\infty}{n\choose k}2^{1-{k\choose2}}$, where $n=\max\{n\in\mathbb{N}:{n\choose k}2^{1-{k\choose2}}<1\}$
How do you prove $\lim_{k\to\infty,k\in\mathbb{N}}{n\choose k}2^{1-{k\choose2}}=1$, where $n=\max\{n\in\mathbb{N}:{n\choose k}2^{1-{k\choose2}}<1\}$? The expression ${n\choose k}2^{1-{k\choose2}}$ ...
- 276
1
vote
1
answer
276
views
What is the expected number of missing random integers?
Consider $n$ numbers randomly generated by independent generators that can produce integers from $0$ to $n$. How many of these integers will be missing on average for large $n$? If $p_{k,n}$ is the ...
- 1,731
1
vote
2
answers
912
views
Multinomial Coefficient Estimates
Hello,
Let $B$ and $n$ be positive integers. Let $p_i \ge 0 $ be such that $\sum_{i=0}^{2B} p_i= 1$.
I am interested in asymptotics (in terms of $B$, $n$, and $p_i$) for the coefficients of
$
(...
- 11
1
vote
1
answer
124
views
Tight estimates for binomial summation
Is there tight estimates for the following logarithmic summation ($\gamma\in(0,1)$)
$$\ln\Bigg(\sum_{t=\frac{n^{}}2-\gamma n^\gamma}^{\frac{n^{}}2+\gamma n^\gamma}\sum_{\ell=\frac{n^{}}2-\gamma n^\...
- 1,826
0
votes
1
answer
322
views
Asymptotic of a certain double sum involving binomial coefficients
Consider sums of the form
$S(n)=\sum^{n}_{m=0}\sum^{m}_{k=1}2^{2k+m+1}{n-m+k+1 \choose 2k+2}{m \choose k}$
I am interested in the asymptotics of $S(n)$ as $n\to \infty$.
More precisely I would ...
- 2,964
0
votes
1
answer
163
views
Terminology and approximation to logarithm of a sum of products of binomial coefficients
Denote $$T(m)=\sum_{1\leq n_m\leq n_{m-1}\leq\dots\leq n_2\leq n_1\leq m}\prod_{i=1}^{m}\binom{n_i}{n_{i+1}}.$$
Is there a name for this kind of summation and is there a good estimate for $\ln T(m)$ ...
- 1,826
0
votes
1
answer
642
views
Asymptotic growth for $\sum_{i=1}^{n-1}(n-i)\binom{k}{i}$ [closed]
For positive integers $n,k$, define $$f(n,k):=\sum_{i=1}^{n-1}(n-i)\binom{k}{i}.$$
What are upper and lower bounds of $f(n,k)$ by simpler terms? (e.g. finding bounds which are not a summation like ...
- 119
0
votes
1
answer
208
views
Local behavior of the Vandermonde convolution
An interesting combinatorial identity is the Vandermonde convolution identity:
$$ \sum_k {n\choose k}{m\choose s-k} = {n+m \choose s},$$
which can be proved by considering the coefficients in $(x+1)^{...
- 5,230
0
votes
2
answers
208
views
Interpolating asymptotic expression for logarithm of middle binomial sums
Define $S(k,2n)=\sum_{i=-k}^k\binom{2n}{n+i}$ at every $k\in\{0,\dots,n\}$.
We know $$\ln(S(\gamma\mbox{ } n^\gamma,2n))\asymp(2n\ln2-\frac12\ln\pi-\frac12\ln n)$$ at $\gamma\rightarrow0$ and $$\ln(S(...
- 1,826
0
votes
2
answers
140
views
Asymptotic bound of a simple alternating binomial sum
I'm a rather inexperienced researcher, I've been stuck on a question for a while. I would like to find the largest $N = f(n)$ that satisfies the following inequality:
$$\sum_{j = 0} ^ n p^{n - j} (-1)...
0
votes
0
answers
191
views
Asymtotic Complexity Analysis using logarithms and binomial coefficients
On page 11 of "Smaller decoding exponents: ball-collision decoding" by Berstein et.al. they have the formula \begin{equation}\lim_{n \rightarrow \infty} \frac{1}{n}\log_{2}\left(\dbinom{k_{1}}{p_{1}}\...
- 101