3
votes
3answers
355 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 ...
2
votes
2answers
316 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 ...
0
votes
1answer
287 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 ...
2
votes
0answers
178 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 ...
4
votes
1answer
233 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 ...
2
votes
0answers
191 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 ...
1
vote
2answers
266 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 $ ...
2
votes
3answers
1k 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 ...
0
votes
0answers
177 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} ...
2
votes
1answer
285 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 ...
2
votes
2answers
879 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 ...
6
votes
2answers
684 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
2answers
532 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 ...
2
votes
1answer
611 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 $n\geq k$, ...
11
votes
5answers
844 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) = ...