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3
votes
3answers
455 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 ...
3
votes
1answer
258 views

Bounded convolutions with binomial coefficients

I need to figure out a nice family of decaying functions such that $\sum_{d=2}^k {k \choose d} f_k(d) \leq 1/k$ and $f_k(d)\geq f_k(d+1)$ How can I figure out what good candidates could be? Any ...
2
votes
2answers
358 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
458 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 ...
3
votes
1answer
326 views

Limit of sum of binomials

I'm trying to calculate the limit for the sum of binomial coefficients: $$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right).$$ Numerically it seems to ...
7
votes
3answers
650 views

Combinatorial identities

I have computational evidence that $$\sum_{k=0}^n \binom{4n+1}{k} \cdot \binom{3n-k}{2n}= 2^{2n+1}\cdot \binom{2n-1}{n}$$ but I cannot prove it. I tried by induction, but it seems hard. Does anyone ...
2
votes
0answers
283 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 ...
0
votes
1answer
274 views

Asymptotic behaviour of Binomial Sum

I am interested in the behaviour of: $\gamma_k=\sum_{i=0}^{k} {n \choose i}$ as n becomes large and where $k$ could potentially be a function of $n$ rather than a constant. One line of attack I can ...
0
votes
1answer
186 views

About the number of the elements of a set related with binomial coefficients

For $N\in\mathbb N$, let $$P_l(N)=\# \{(n,m)|0\le n\le N, 0\le m\le n,\binom{n}{m}\not\equiv 0 \mod l\}.$$ Suppose that $\binom{n}{0}=1$ for $n\ge 0$ and that $\# S$ represents the number of the ...
9
votes
2answers
1k views

Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$

I found the following formula in a book without any proof: $$\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}.$$ This does not seem to follow immediately from the basic ...
2
votes
3answers
688 views

An identity involving sum of probably binomial coefficients

How could I prove that $$\sum _{m=v}^n \left(\left(\prod _{k=v}^{m-1} \frac{k^2}{m^2-k^2}\right)\left(\prod _{k=m+1}^n \frac{k^2}{k^2-m^2}\right)(-1)^{m-v}\right)=1$$ or, simplified, $$\sum _{m=v}^n ...
0
votes
1answer
134 views

Congruences for generalized Franel numbers

Let us define generalized Franel numbers $f^{(m)}_n$ through recurrence relations: $f^{(1)}_n=1$ for all $n$, and $$f^{(m)}_n=\sum\limits_{k=0}^n\binom{n}{k}^3f^{(m-1)}_k.$$ In fact ...
2
votes
1answer
169 views

Congruence for the Apery Numbers

Is it true that $$A_n\equiv (-1)^n\;\;(\mathrm{mod}\;3)\;\;?$$ Here $A_n$ is the Apery number: $$A_n=\sum\limits_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2.$$ What is known about congruence properties ...
4
votes
1answer
352 views

Product of central binomial coefficients

I have a question about an equality involving products of central binomial coefficients. If $x_1,...,x_n$ and $y_1,...,y_n$ are positive integers, with $\sum_i x_i = \sum_i y_i$ and $$ ...
4
votes
0answers
258 views

Recurrence relation for trinomial Apery numbers

It is well known (Beukers 1987) that the Apery numbers $$A_n\equiv A_n^{(2)}=\sum\limits_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2$$ satisfy the fancy recurrence relation ...
10
votes
2answers
619 views

Why are negative sets multisets? (Reference request)

It is easy to establish that $$ \left(\!\!{n\choose k}\!\!\right)=(-1)^k{-n \choose k}, $$ where the symbol on the left-hand-side counts the number of multisets of $k$ elements from $n$. On the ...
4
votes
0answers
85 views

alternating sum with Barnes G functions

Let $B(n)=(n-2)!(n-3)!\cdots 1!$ denote the Barnes G-function. I am pretty sure that $$ \sum_{m=0}^{k^2-1} (-1)^m\binom{k^2-1}m \frac{G(k+n-m+1)}{G(n-m+1)G(k+1)(k^2)!} = n-2k^2-2k $$ when $k$ is ...
4
votes
1answer
323 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
261 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
135 views

Resources on Wolstenholme's theorem

In Wikipedia entry on Wolstenholme's theorem, it says ...
2
votes
0answers
454 views

Another identity involving sums of (alternating) binomial coefficients.

I have derived two different solutions to the same problem involving computing the expected time to find $k$ swaps when collecting coupons. However to me the two sums, although apparently numerically ...
3
votes
1answer
729 views

Estimate on sum of squares of multinomial coefficients

I am interested in approximating the sum of the squares of the multinomial coefficients, i.e. $a_\ell^p := \sum_{k_0+\ldots+k_p = \ell} (\frac{\ell!}{k_0! \ldots k_p!})^2$ or more general, ...
18
votes
1answer
963 views

Solutions to $\binom{n}{5} = 2 \binom{m}{5}$

In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says: On National Public Radio, the Weekend Edition program posed the following probability problem: Given a certain number of ...
5
votes
1answer
456 views

Elementary proof for identity involving sums of binomials

Is there an elementary proof of this identity? $$n + 1 - \sum_{k=1}^{n} k^{k-1} \binom{n}{k} \frac{(n-k)^{n+1-k}}{n^{n}} =1 + \sum_{k=1}^n \frac{n!}{(n-k)!n^k}\;?$$ The term on the right is the ...
1
vote
0answers
93 views

binomial transform, Hurwitz zeta function

For $j,n\in\mathbb Z_+$, let $$ L_{j,n}^{(t)}= \sum_{m=0}^{n} \Bigl(-\frac 12\Bigr)^{n-m}{n\choose m}{m+j+1\choose m+1} \left( \frac {1}{t+\frac 12}\right)^{m+j+2} $$ and $$ L_{j,n} ...
3
votes
2answers
543 views

Sum involving binomial coefficients

I have the following sum $\sum_{j=1}^K {K \choose j} (-1)^{j+1}/j$. Now I can write this as the integral $\int_{-1}^0 \frac{(1+x)^K - 1}{x} dx$. However, I wonder whether there is a closed form ...
3
votes
2answers
204 views

A sum related to the Johnson association scheme

Hi everyone, In the process of studying a problem in the Johnson association scheme I came across the following sum: $$\sum_{k\geq 0}(-1)^k\binom{n}{k}\binom{a-k}{a-b}\binom{c+k}{b}.$$ All the ...
1
vote
2answers
754 views

An identity involving a sum of binomial coefficients

I am moving through a classic paper (On Average Height of Planted Plane Trees by Knuth, de Bruijn and Rice, 1972), and I would like to trade a weaker result for simpler mathematical tools, because my ...
5
votes
2answers
534 views

Interpolating a sum of binomial coefficients using a sin function

While studying a problem about orthogonal polynomials I encountered the following expressions \begin{equation} f(n)=\sum_{k=0}^{n}(-1)^k\binom{n+k}{2k} \frac{1}{k+1}\binom{2k}{k} \end{equation} and ...
0
votes
0answers
729 views

Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...
1
vote
2answers
443 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 $ ...
0
votes
1answer
300 views

Sum involving integer compositions and binomial coefficients

I came across an identity involving binomial coefficients. I'm not sure if I'm looking at the identity the wrong way but I am not aware if this identity is known and if there is an (easy) proof for ...
4
votes
0answers
266 views

A coincidence concerning Fermat primes, binomial sums, and eta quotients?

Let $w_k$ be a primitive k th root of unity, where k is a power of 2. In response to a question, Robert Israel gave the solution, $$\sum_{n=0}^\infty \frac{(-1)^n}{\binom{kn}{kn/2}} = ...
4
votes
1answer
2k views

sum calculation

I would like to calculate, or bound from above, the following sum $$ \sum_{i=0}^n(n-2i)^p{p \choose i}, $$ here $p\geq 2$. Any references are very welcome. Thank you.
2
votes
3answers
2k 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
179 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} ...
3
votes
1answer
361 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
1k 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 ...
30
votes
3answers
2k views

A limit involving binomial coefficients?

Experimentation suggests the limit $$\lim_{n\rightarrow\infty} (-1)^n\sum_{k=1}^n(-1)^k{n\choose k}^{-1/k}=\frac{1}{2}\ .$$ Does somebody have an idea for (a start of) a proof? Added: There seem to ...
7
votes
0answers
327 views

Polynomials with presumably positive coefficients

The $q$-Pochhammer symbol $(q) _ 0:=1$ and $(q) _ n:=\prod _ {j=1}^n (1-q^j)$ for $n > 0$ is clearly a polynomial in $q$ which has both positive and negative coefficients when $n>0$. The ...
6
votes
3answers
519 views

Is there a closed formula for the generating function of some trinomial coefficients?

We learn in calculus how to obtain a sum of binomial coefficients $\frac{(2d)!}{(d!)^2}$ in terms of a generating function $\sum_{d \geq 0} \frac{(2d)!}{(d!)^2} x^d$ by the Taylor series of ...
40
votes
2answers
3k views

Alternating sum of square roots of binomial coefficients

Let $$ c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}. $$ It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...
8
votes
0answers
535 views

Does anyone know this sequence of polynomials?

A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$. \begin{align} P_{0,0}&=1\\ \text{for $n\geq ...
1
vote
1answer
337 views

Limit of an infinite series, related to (generalised Newton's) binomial expansion

How to calculate the infinite sum of the following series, related to binomial expansion for rational number, $r$: ...
6
votes
1answer
682 views

Sums of binomials with even coefficients

While looking for a closed form of a expression I worked myself to a formula that resembles the Vandermonde convolution, but is summed over even binomial coefficients only. ...
0
votes
2answers
1k views

non negative integer solutions : Diophantine Equations [closed]

I want to know the exact number of non-negative integer solutions of $a_1 + 2a_2 + \ldots + k \cdot a_k = n$. I know that it is the co-efficient of $x^n$ in $(1 - x^{a_1})^{-1} \cdot (1 - ...
1
vote
1answer
1k views

How to calculate such sum of product of binomial coefficient?

..I wonder if the following formula can be calculated? $ \sum_{k=0}^m {m \choose k} {2k \choose n} $
7
votes
3answers
867 views

Binomial coefficient in Andrews' partition book

First of all, i think MathOverflow is a very great community to discuss math, either basic or advanced, and i'm glad to participate here. It's my first post, so i'm sorry if i did anything wrong, and ...
0
votes
0answers
178 views

$q$-binomials in terms of ordinary binomials

Is there a known identity that expresses the $q$-binomial coefficients (also known as Gaussian binomial coefficients) in terms of the (ordinary) binomial coefficients?
4
votes
1answer
508 views

What is the degree of a symmetric boolean function?

(previous title " Zero sum of binomials coefficients - a stronger version ") This is a stronger version of another question. Is there an $N\in \mathbb N$ and a sequence of non-constant functions $ ...