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36
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 ...
36
votes
0answers
2k views

How to prove this polynomial always has integer values at all integers?

Let $m$ be any positive integer. $$ P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}. $$ Question: $P_m(x)$ always ...
33
votes
4answers
3k views

Integer-valued factorial ratios

This historical question recalls Pafnuty Chebyshev's estimates for the prime distribution function. In his derivation Chebyshev used the factorial ratio sequence $$ ...
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 ...
29
votes
3answers
2k views

A binomial generalization of the FLT: Bombieri's Napkin Problem

This is an extract from Apéry's biography (which some of the people have already enjoyed in this answer). During a mathematician's dinner in Kingston, Canada, in 1979, the conversation turned ...
23
votes
1answer
584 views

integers which are sums of binomial coefficients: $\sum_i {n \choose k_i}$

Let $n$ be an integer. For $S$ a subset of $\{0,\dots,n\}$, define $$m(S) = \sum_{k \in S} {n \choose k}.$$ Let $M_n$ be the set of integers of the form $m(S)$ for all sets $S \subset \{0,\dots,n\}$. ...
18
votes
3answers
2k views

Need help proving that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$

Hello. I have been trying very hard to show that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$ and could not quite get anywhere. This inequality has been verified by computer ...
18
votes
1answer
919 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 ...
17
votes
0answers
496 views

Does $\binom{2n}{n} \equiv 2 \pmod p$ ever hold?

Well, the title does not tell the whole story; the complete question is: Are there any primes of the form $p=2n(n-1)+1$, with integer $n\ge 1$, such that $$ \binom{2n}{n} \equiv 2\pmod p ? $$ ...
15
votes
1answer
899 views

Prove that expression is integer

Numerical experiments suggest that $\binom{2m}{m + k}\cdot\frac{3m - 1 - 2k^2}{2m - 1}$ is integer for all $-m \le k\le +m$. It means that expression evaluation could be implemented very efficiently, ...
15
votes
1answer
1k views

How to prove that the following double sum is always an integer?

I have verified the following double sum is always an integer for $s$ up to $1000$ via Maple. But I can not prove it. Proofs, hints, or references are all welcome. Thanks! ...
14
votes
2answers
2k views

Binomial supercongruences: is there any reason for them?

One of the recent questions, in fact the answer to it, reminded me about the binomial sequence $$ a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2, \qquad n=0,1,2,\dots, $$ of the Apéry ...
11
votes
5answers
881 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) = ...
10
votes
4answers
1k views

Rational congruence of binomial coefficient matrices

Skip Garibaldi asks if there is an elementary proof of the following fact that "accidentally" fell out of some high-powered machinery he was working on. Say that two matrices $A$ and $B$ over the ...
10
votes
1answer
339 views

Equation with $q$-binomial coefficients

Let $d\ge2$, and let $q$ be a power of a prime. As usual, define $N(d,q)=\sum_{k=0}^d{d\choose k}_q$. I wonder if there are $d$ and $q$ as above such that $1+N(d,q)=q^{d+1}$. (If the answer is ...
10
votes
1answer
632 views

Simplest form for sum of Binomial Expressions

How difficult is the problem of reducing the number of terms in a sum of binomial expressions? Formally: Given $a_1, a_2, a_3, … a_n$, and $b_1, b_2, b_3, ... , b_n$, where $a_i, b_i \in \mathbb{Z}$, ...
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 ...
9
votes
3answers
2k views

Showing a matrix is negative definite [formerly Showing a sum is always positive]

For each $d$, I have a matrix $M$ with values $$ M_{ij} = \begin{cases} \frac{4ij}{d} - \binom{2d}{d} & i \neq j & \\\\ \frac{4i^2}{d} - \binom{2d}{d} - ...
9
votes
7answers
4k views

Lower bound for sum of binomial coefficients?

Hi! I'm new here. It would be awesome if someone knows a good answer. Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case ...
9
votes
2answers
852 views

What is $\sum_{i=0}^{n}\binom{n}{i}^3$?

We know that $$\sum_{i=0}^{n}\binom{n}{i}=2^n$$ and that $$\sum_{i=0}^{n}\binom{n}{i}^2= \binom{2n}{n}$$ what about $$\sum_{i=0}^{n}\binom{n}{i}^3$$ ?
9
votes
3answers
1k views

Solving a general two-term combinatorial recurrence relation

What is known about explicit (not necessarily closed-form) solutions to the recurrence $$R^n_k= (\alpha n) R^{n-1}_k + (\alpha' n + \beta' k) R^{n-1} _{k-1},$$ with initial condition $R_0^0 = 1$ and ...
8
votes
2answers
569 views

Zero sum of binomial coefficients

Is there a function $p:\mathbb N\to \{ 1,-1 \} $ and a fixed $N\in \mathbb N$ such that for every $n \geq N$ we get: $\sum _{i=0} ^{n} p(i)\binom {n}{i}=0$ ? Obviously $p(i)=(-1)^i$ works for ...
8
votes
1answer
483 views

What is $\lim_{n\to\infty} \displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?$

What is $$\lim_{n\to\infty} \displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?$$
7
votes
3answers
818 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 ...
7
votes
1answer
364 views

Quest for a human proof of a $q-$binomial identity

Let $$f(n,k) = \sum\limits_{j = - k}^k {{{( - 1)}^{k - j}}} \binom{n-j}{k-j}\binom{n+j}{k+j}.$$ Then $f(n,k)=\binom{n}{k}$ because it satisfies $f(n,k)=f(n-1,k)+f(n-1,k-1)$ and the obvious ...
7
votes
0answers
314 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 ...
7
votes
0answers
724 views

Inverse of a matrix with binomial coefficients

Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is ...
6
votes
3answers
920 views

Multiplicative Convolution for Binomial Coefficients

I know Vandermonde's convolution for binomial coefficients: $$\sum_{j=0...k} \binom{n}{j} \binom{m}{k-j} = \binom{n+m}{k}$$ Is there a similar multiplicative convolution? More precisely, is there a ...
6
votes
1answer
641 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. ...
6
votes
1answer
792 views

Divisibility of a binomial coefficient by $p^2$ — current status

While skimming the book Concrete Mathematics, (edit: first edition) I came across the following problem, which is listed there as a Research Problem: (Chapter 5, Exercise 96) Is ${2n \choose n}$ ...
6
votes
3answers
472 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 ...
6
votes
2answers
545 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 ...
6
votes
2answers
729 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
480 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 Wikipedia ...
6
votes
0answers
465 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 $0\leq k\leq n/2$. ...
5
votes
3answers
612 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 ...
5
votes
6answers
614 views

Binomial coefficient identity

It seems to be nontrivial (to me) to show that the following identity holds: $$ \binom {m+n}{n} \sum_{k=0}^m \binom {m}{k} \frac {n(-1)^k}{n+k} = 1. $$ This quantity is related to the volume of the ...
5
votes
2answers
494 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 ...
5
votes
1answer
434 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 ...
5
votes
4answers
482 views

Maximum value of the binomial coefficient as a polynomial

What is the maximum (absolute) value of the binomial coefficient $\begin{pmatrix}x \\ k\end{pmatrix} = \frac{1}{k!}x(x-1)(x-2)\dotsb(x-k+1)$ for real $x$ in the interval $0 \leq x \leq k-1$?
5
votes
2answers
212 views

Combinatorial identity and Fuss-Catalan numbers

I would like to show that $$ \lim_{N\to\infty}\frac{1}{N^{np+1}}\frac1{p!}\sum_{j=0}^{p-1}(-1)^j\binom{p-1}{j} \left(\frac{\Gamma(N+p-j)}{\Gamma(N-j)}\right)^{n+1} =\frac1{np+1}\binom{(n+1)p}{p}, $$ ...
5
votes
0answers
112 views

An inequality concerning non-negative integer matrices with constant row and column sums

[I posted this question on math.stackexchange a few weeks back, but no luck there so far: ...
4
votes
1answer
338 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
1answer
496 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 $ ...
4
votes
1answer
287 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 ...
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.
4
votes
0answers
149 views

How find this binomial-coefficients sum $\sum_{k_{1}+k_{2}+\cdots+k_{d}=n}\binom{n}{k_{1},k_{2},\cdots,k_{d}}^2$ [duplicate]

Assmue that $d$ is give postive integer numbers,and $$(x_{1}+x_{2}+\cdots+x_{d})^n=\sum_{k_{1}+\cdots+k_{d}=n} \binom{n}{k_{1},k_{2},\cdots,k_{d}}x^{k_{1}}_{1}x^{k_{2}}_{2}\cdots ...
4
votes
0answers
389 views

Double sum involving binomial coefficients

I came across a sum of binomial coefficients while trying to solve a problem involving $SU(2)$ group integrals. I am not able to solve it, nor I found a similar identity in the literature. I would ...
4
votes
0answers
250 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 ...
4
votes
0answers
81 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 ...