The binomial-coefficients tag has no usage guidance, but it has a tag wiki.

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### 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 ...

**16**

votes

**1**answer

644 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 ...

**18**

votes

**1**answer

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 $a_i, b_i \in \mathbb{Z}$, ...

**10**

votes

**2**answers

618 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 ...

**8**

votes

**0**answers

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 ...

**4**

votes

**3**answers

385 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

**0**answers

81 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 ...

**1**

vote

**1**answer

213 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 ...

**23**

votes

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3k views

### Is the sum $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0?$

I am trying to prove $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$. This inequality has been verified by computer for $k\le40$.
Some clues that might work (kindly provided by ...

**5**

votes

**2**answers

380 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 ...

**2**

votes

**0**answers

84 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 ...

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votes

**6**answers

861 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 ...

**13**

votes

**1**answer

370 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 ...

**6**

votes

**3**answers

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 ...

**2**

votes

**1**answer

145 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

**0**answers

91 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 ...

**3**

votes

**1**answer

151 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 ...

**1**

vote

**1**answer

52 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 ...

**3**

votes

**1**answer

140 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

**0**answers

53 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 ...

**2**

votes

**1**answer

208 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 - ...

**6**

votes

**1**answer

585 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 ...

**1**

vote

**1**answer

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

**1**answer

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 ...

**42**

votes

**1**answer

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 ...

**4**

votes

**3**answers

162 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 ...

**3**

votes

**1**answer

151 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

**0**answers

147 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 ...

**0**

votes

**2**answers

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 - ...

**15**

votes

**1**answer

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

**2**answers

244 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 ...

**3**

votes

**2**answers

153 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

**1**answer

372 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 ...

**0**

votes

**1**answer

289 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 ...

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votes

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416 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

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323 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 ...

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votes

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305 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

**1**answer

238 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 ...

**0**

votes

**1**answer

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 ...

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votes

**0**answers

137 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: ...

**7**

votes

**1**answer

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**

votes

**2**answers

865 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

**1**answer

285 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 } ...

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votes

**2**answers

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 ...

**2**

votes

**1**answer

162 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 ...

**1**

vote

**0**answers

64 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

**1**answer

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!
...

**2**

votes

**1**answer

233 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

**0**answers

150 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 ...

**18**

votes

**0**answers

580 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
$$ \binom{2n}{n} \equiv 2\pmod p ? $$
...