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0
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0answers
24 views

A sum involving binomial coefficients and its evaluation using the Gamma function [migrated]

Does anyone know how to prove (or a reference for) the following identity for positive integers $r$: $$\sum_{i=0}^r (-1)^i{r\choose i}\frac{1}{ir+1}= \frac{\Gamma(1+1/r)\Gamma(r+1)}{\Gamma(r+1+1/r)}$$
1
vote
1answer
118 views

estimating binomial coefficients

There is a beautiful paper on the arXiv by Andrew Suk containing an asymptotic result about the Erdös-Szekeres convex polygon problem. I am struggling with one of the estimates he makes on page 4. He ...
1
vote
1answer
201 views

Simplify a equation

I have a problem simplifying the summation here: $$ \sum_{x=0}^{n}\sum_{y=0}^{x} {n\choose{x}} {x\choose{y}} y!(x-y)! $$ The last three terms can be simplified to x!, so the current summation ...
15
votes
1answer
787 views

Are there good bounds on binomial coefficients?

Motivated by the central limit theorem, one expects that $$\binom{n}{k} \approx \frac{2^n}{\sqrt{\pi n/2}} \exp\left(-\frac{(k-n/2)^2}{n/2}\right).$$ Computations suggest that the ratio of the two ...
5
votes
1answer
216 views

Estimate of incomplete binomial integral

Let $0\le k \le n$. Prove that $$ n\binom{n}{k}\int_{0}^{\frac{k}{n+1}}t^k(1-t)^{n-k}\,dt \le 1/2. $$ As far as I know 1) it is proved for $\frac{k}{n+1}\le 1/2$ and 2) not proved for $1/2 <\frac{...
4
votes
3answers
397 views

On a theorem of Hensel about congruence of binomial coefficient

In the paper Binomial coefficients modulo prime powers, Andrew Granville stated the following theorem: Let $n, m$ and $r=n-m$ be three given positive integer and $p^k$ is the exact power of $p$ ...
1
vote
0answers
87 views

Sum of products of binomial coefficients

I have been trying to check that two things are equal for a while now, Mathematica appears to say that they are but I can't for the life of me figure out how to show it (without just saying 'computer ...
2
votes
0answers
88 views

A combinatorial sum involving ratios of binomials [closed]

Can anyone suggest how to prove the following (for $k \le n$): $$\sum \limits_{s=0}^N \frac{\binom{n}{k} \binom{N-n}{s-k} }{\binom{N}{s}} = \frac{N+1}{n+1}$$ I am assuming it to be true, and ...
15
votes
1answer
395 views

A combinatorial identity involving generalized harmonic numbers

The $n$-th harmonic number is defined as $$ H_n=\sum_{k=1}^{n}\frac{1}{k}, $$ and the generalized harmonic numbers are defined by $$ H_{n}^{(r)}=\sum_{k=1}^{n}\frac{1}{k^r}. $$ Recently, I have found ...
2
votes
1answer
152 views

Alternating sign binomial identity [closed]

I recently noticed that for a triple of integers $k \geq 2$, $k \geq m \geq t \geq 1$, the following identity seems to hold $\sum_{j=0}^{m-t} (-1)^{m-t-j}{k \choose j}{m-1-j \choose t-1}={k-t \choose ...
0
votes
0answers
110 views

Partial sum of binomial coefficients

For some integer $z \ge 2$ and large integer $n$ and $ t=\lceil \log n\rceil $, what is an approximate value for the following partial binomial sum? $$ \sum_{i=0}^{n-t} \binom{n}{i}z^i .$$ Another ...
5
votes
2answers
385 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 ...
1
vote
1answer
57 views

Summation of multinominal coefficients with extra bounds on summation indices

My question is related to the sum \begin{equation} S(n,N) = \sum_{k_1+k_2+...+k_N=n}\frac{n!}{(k_1!)\cdot(k_2!)\cdot...\cdot(k_N!)} = N^n, \end{equation} which is comes from the multinomial theorem:...
3
votes
1answer
142 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 ...
3
votes
0answers
55 views

How to prove that $\sum_{i=0}^n\frac{(a;q)_i}{(q;q)_i}\frac{(b;q)_{n-i}}{(q;q)_{n-i}}a^{n-i}=\frac{(ab;q)_n}{(q;q)_n}$?

By Cauchy identity, $${}_1\phi_0(a;—;q,z)=\sum_{n\geq0}\frac{(a;q)_n}{(q;q)_n}z^n=\frac{(az;q)_{\infty}}{(z;q)_\infty},\quad|z|<1,|q|<1,$$ we can obtain the $q-$analogue of $(1-z)^{-a}(1-z)^{-b}=...
2
votes
1answer
216 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 - ...
1
vote
1answer
155 views

limit and combinatorics

Given $x \in (0,\frac{1}{2})$ and $y \in (0,\frac{1}{2}]$, what is the value of the following limit: $\lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}|?$ When $...
3
votes
1answer
155 views

Effective Realization of GCD of middle binomials?

So, it is well-known that $$ \gcd \left(\binom{m}{k}\mid 1\leq k\lt m \right) = e^{\Lambda(m)}$$ which can incidentally be sparsified for prime $p$ $$ \gcd \left(\binom{p^{r+1}}{1},\binom{p^{r+1}}{p^...
4
votes
3answers
163 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!)^...
3
votes
1answer
167 views

Closed Form Expression for Nested Series Summation?

Just wandering if there are any criteria that can decide whether a finite series summation has closed form or not. for example, In the following nested summation, $n$ is some even integer that will be ...
2
votes
0answers
149 views

binomial coefficients and irrationals

The following, probably either currently impossible to deal with, or having a negative solution, arose from an ergodic theory question, presumably itself currently intractible. I am not a number ...
15
votes
1answer
1k 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, ...
2
votes
2answers
249 views

binomial/factorial identity mod p

In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result. Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...
42
votes
1answer
3k 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 ...
3
votes
2answers
156 views

Closed form for binomial coeff sum

As part of a proof in finite group theory, I'm looking for a closed form for the expression $$\sum_{i=j+1}^{n} \binom{\binom{i}{j}}{2}$$ Any help - especially with reference or proof - would be ...
9
votes
1answer
389 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 ...
3
votes
3answers
423 views

A question about summation formula involving binomial coefficient

In Table of Integrals, Series, and Products. Seventh Edition. I.S. Gradshteyn and I.M. Ryzhik, there is 0.154.3 $$ \sum_{k=0}^N (-1)^k {N \choose k} k^{n-1} =0, N \geq n \geq 1; 0^0 ≡ 1 $$ 0.154.4 $...
1
vote
2answers
498 views

Partial Sum of the Binomial Theorem [duplicate]

The binomial theorem states $\sum_{k=0}^nC_n^kr^k=(1+r)^n$. I am interested in the function \begin{equation} \sum_{k=0}^mC_n^kr^k, \quad m<n \end{equation} for fixed $n$ and $r$, and both $m$ and $...
5
votes
2answers
325 views

Combinatorial identity and Fuss-Catalan numbers

I would like to show that $$ \lim_{N\to\infty}\frac{1}{N^{np+1}}\frac1{p!}\sum_{j=0}^{p-1}(-1)^j\binom{p-1}{j} \left(\frac{\Gamma(N+p-j)}{\Gamma(N-j)}\right)^{n+1} =\frac1{np+1}\binom{(n+1)p}{p}, $$ ...
1
vote
1answer
247 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 ...
5
votes
0answers
142 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: http://math.stackexchange.com/questions/1095659/an-inequality-concerning-non-negative-integer-matrices-with-...
7
votes
1answer
378 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 ...
9
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2answers
869 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 $$\sum_{i=0}^{n}\binom{n}{i}^3$$ ?
2
votes
1answer
289 views

$n^3 | \sum_{i=1}^{n-1}\binom{n}{i}^2$ => $n | \sum_{i=1}^{n-1}\binom{n}{i}$?

For $n\in \mathbf{N}$ is $$n^3 \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}^2=\binom{n}{1}^2+\cdots +\binom{n}{n-1}^2$$ impling $$n \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}=\binom{n}{1}+\...
2
votes
1answer
168 views

Are binomial coefficients $F_1$ analogs of $q$-binomial coefficients?

This is a mostly philosophical question. Is it fair to think of usual binomial coefficients and their identities as an $F_1$ case of $q$-binomial coefficients and identities? Here $F_1$ is the field ...
7
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6answers
892 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 ...
1
vote
0answers
66 views

closed form for a series with binomials and primes

does the series $\sum_{n=0}^\infty p^n \binom{x}{p^n}$ have a closed form ? ($p$ prime) this is a special case of $\sum_{n=0}^\infty p^n \left(\sum_{k=p^n}^{p^{n+1}-1}a_k\binom{x}{k}\right)$ with the ...
17
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! $$\sum_{m=s}^{2s}\sum_{k=0}^{...
19
votes
1answer
1k 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 $a_i, b_i \in \mathbb{Z}$, ...
2
votes
1answer
249 views

Upper bound of sum of binomial coefficients

I am looking for an upper bound - up to constant factor - for: $\sum_{k=t}^{t+l} {n \choose k} \cdot 2^{-n}$ where: The values of $t$ are between: $\frac{n}2+\sqrt{n} \leq t \leq \frac{9n}{10}$. (...
4
votes
0answers
154 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 $$(x_{1}+x_{2}+\cdots+x_{d})^n=\sum_{k_{1}+\cdots+k_{d}=n} \binom{n}{k_{1},k_{2},\cdots,k_{d}}x^{k_{1}}_{1}x^{k_{2}}_{2}\cdots x^{k_{d}}_{d}$$ ...
18
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0answers
606 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 $$ \binom{2n}{n} \equiv 2\pmod p ? $$ ...
1
vote
2answers
310 views

The zeros of alternating sign, binomial coefficient polynomials

I have a question regarding the zeros of the following polynomial, based on partial rows of pascal's triangle, $$f(x)=\sum_{k=a}^n\binom{n}{k}(-1)^{k}x^k,$$ where $a,n\in \mathbb{Z}^+,n>a.$ ...
0
votes
1answer
290 views

Maximize combinatorial sum for boolean function

I am trying to maximize the function $$ S(f)=\sum_{j=0}^{n-\frac{n-1}{t}}(-1)^j{n-\frac{n-1}{t}\choose{j}}\sum_{i=0}^{\frac{n-1}{t}}(-1)^{f(i-j)}(t-1)^i{\frac{n-1}{t}\choose{i}} $$ for a function $f:\...
1
vote
0answers
163 views

Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$ \begin{array}{cccccccccc} 1 & = & K_{1}\tbinom{...
6
votes
4answers
704 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 $0 \leq x \leq k-1$?
23
votes
1answer
603 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 $S \subset \{0,\dots,n\}$. ...
2
votes
1answer
190 views

Combinatorial sum (Author and generalization?)

In a book I have met one interesting equation (without reference): $$\frac{m!}{n!}\sum_{i=0}^n(-1)^i{n\choose{i}}{x+m+n-i\choose{m}}=\begin{cases} x+n+1,\, if \,m=n+1 \\ 1,\, if \,m=n \\ ...
0
votes
0answers
111 views

Estimating when does a certain binomial sum exceed an upper bound

Given a fixed integer $n > 0$ and $0 \le m \le n$ let us define the numbers $$f_{n,m} = \sum_{i=\lfloor m/2 \rfloor}^m {n-2i \choose n - m -i}{i+1 \choose m - i +1}.$$ For example $f_{n,0} = 1,f_{...
1
vote
1answer
223 views

Polynomial convex coefficients

Assume we have an arbitrary high order polynomial $$f(L)=1-L\theta_1-L^2\theta_2-L^3\theta_3-...-L^N\theta_N$$ and we know all roots of this polynomial site outside the unit circle. It is obvious that ...