1
vote
0answers
10 views

Approximation method for combinations

I have the expression: $$\frac{{D+\frac{k}{\theta}}\choose D-z}{{1+m}\choose{D-z}}$$ In the above expression, $m$, $k$, $\theta$, and $D$ are constants. What is the approximation of the above ...
1
vote
0answers
9 views

Simplifcation of the Function Required

I want to simplify the following expression $$ \sum_{x=0}^{D+m} \frac {x!} {(n+1)_x} (\frac {\theta_{eff}} {\mu})^x {D+m \choose x} $$ The parameters $D, m, n, \theta_{eff}, \mu, $ are constants. ...
1
vote
2answers
49 views

Is $n\binom{\epsilon n}{t}>t\binom{n}{t}$ for large $n$ and fixed $\epsilon$ and $t$

Let $\epsilon$ and $t$ be fixed numbers with $t$ and integer. I came across the following inequality in a counting problem. $$n\binom{\epsilon n}{t}>t\binom{n}{t}.$$ I want to show that for $n$ ...
0
votes
1answer
50 views

Approximation of $\frac{1}{2^{N}} \binom{N}{\frac{1}{2}(N + s)} $ for big N

I have the following function that I have to approximate for big N and I really have no clue what to do - I hope that anyone can help me out: $$P_N(s) = \begin{cases} \frac{1}{2^{N}} ...
1
vote
0answers
40 views

Approximating the probability that a range bounds a given number, with very large numbers

Let $m$ numbers be chosen uniformly from $0,\dots,n-1$ without replacement and then sorted in ascending order as $\ell_0,\dots,\ell_{m-1}$. Let there be $b,e,x$ such that $0 \le b \le e \le m$ and $0 ...
6
votes
1answer
133 views

Simple approximation to a sum involving Stirling numbers?

I have also posted this question at http://mathoverflow.net/questions/141552/simple-approximation-to-a-sum-involving-stirling-numbers#141552. I have an exact answer to a problem, which is the ...
2
votes
0answers
91 views

Good upper bound on a binomial sum

What is a good upper bound on the following binomial sum: $$\sum_{i,j< \frac{m}{n}} {m \choose i}{m-i \choose j} z^{i+j}$$ where $z = \frac{1}{n-2}$?
2
votes
0answers
84 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 ...
3
votes
0answers
40 views

Bound on Permutations [duplicate]

I am trying to prove the following inequality, $$n^{(l)} = n(n-1)\cdots(n-l+1) \geq \frac{n^l}{e}\quad\text{ for }\quad 2 \leq l \leq \sqrt{n}\;.$$ So my approach is to observe that $n^{(l)} = ...
3
votes
3answers
177 views

$t=\frac{30^{65}-29^{65}}{30^{64}-29^{64}}$, find the closest pair of integers, a and b, such that, $a \lt t \lt b$.

$t=\frac{30^{65}-29^{65}}{30^{64}-29^{64}}$ find the closest pair of integers, a and b, such that, $a \lt t \lt b$. $30=1+29$ $(1+29)^{65}=(1+29)(1+29)^{64}$
0
votes
1answer
100 views

Counting Weak Compositions and Approximating Alternating Sum

I have the following problem: "Suppose you have a universe of $N$ distinct objects, and you observe $k$ of them, possibly with repetition. The order in which the objects are observed does not matter. ...
0
votes
0answers
110 views

Calculation of sum

I am wondering if it is possible to calculate or approximate the following sum $$ \sum_{k=0}^l\frac{(l-2k)^p(2l+k(k-1))l^{k-1}}{(k+3)(k+2)} $$here $p \geq 2$. Thank you.
4
votes
1answer
101 views

Proof about binomial coefficient

I today see a approximated equation, when $n \ll u $: $$\log {u \choose n} \approx n \Big(\log \frac{u}{n} + 1.44\Big)$$ I would like to know how to prove it.
6
votes
2answers
405 views

Elementary bound of binomial coefficient

I'm working my way through an Erdős paper from the sixties and some of the elementary bounds he claims seem to be just beyond my reach. The expression looks horrendous but maybe there is a clever ...
1
vote
0answers
285 views

Calculation of a 'double' sum

Let $n \in N$ and $q\geq 2$. I am trying to calculate the following sum: $$ \sum_{i=0}^{\sqrt n/2}\sum_{j= i \sqrt n }^{(i+1)\sqrt n}\frac{(-1)^q2^q(\frac{n}{2}-j)^q}{(n-j)!j!} $$ Any help will be ...
4
votes
1answer
198 views

Evaluating a limit involving binomial coefficients.

If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer: $$\lim_{n\rightarrow ...
1
vote
1answer
73 views

Bound for sum with geometric progression

Let $n_i$, $i=1,\ldots,m+1$ be nonnegative natural numbers, sum of which $\sum_{i=1}^{m+1}n_i=N$. I woul like to find an upper bound for the following$$ \sum_{i=1}^{m+1}\frac{\sqrt n_i}{2^{i-1}}$$
8
votes
4answers
2k views

Find bound for sum of square roots

Let $a_1,...,a_n$ be real numbers, such that $a_1+...+a_n=A$. What can we say about $\sqrt{a_1}+...+\sqrt{a_n}$? I would like to bound from above thus sum in terms of $A$.
4
votes
2answers
374 views

computation of the sum

I am having trouble to compute the following sum: $$ \sum_{k=0}^n(n-2k)^p \frac{{n \choose k}{2m-n \choose m-k}}{{2m \choose m}} $$ Here $p\geq 2$. To simplify the question, we can even assume that ...
3
votes
2answers
302 views

Asymptotic number of unlabeled graphs

A rather tight lower bound $c(n)$ of the number of unlabeled digraphs of order $n$ (loops allowed) seems to be $$c(n) = 2^{n^2}/n!$$ because there are $2^{n^2}$ labeled graphs, almost all of them ...