The binomial-coefficients tag has no wiki summary.

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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
...

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### 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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### Combinatorial identity and Fuss-Catalan numbers

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

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### 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 ...

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### 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: ...

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### 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 ...

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

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

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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### 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 ...

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### 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 ...

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### 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 ...

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### 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!
...

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

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### bounds on a series with binomial coefficients

I have the following series
$\sum\limits_{l=1}^n {n\choose l} \alpha^{\beta^l}$
where $\alpha > 0$and $0 \leq \beta \leq 1$.
Can anybody guide me how I can evaluate it or find some tight upper ...

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### 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: $\frac{n}2+\sqrt{n} \leq t \leq \frac{9n}{10}$. ...

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### 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 ...

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

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### 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 $a,n\in \mathbb{Z}^+,n>a.$
...

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### 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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### 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 & = & ...

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### 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 $0 \leq x \leq k-1$?

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### 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 $S \subset \{0,\dots,n\}$. ...

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### 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
\\
...

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### 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} = ...

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### 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 ...

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

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### 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 ...

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### An extrasensory perception strategy :-)

I asked this question at MSE some months ago
but I received only partial answers, so I put it here. The following sounds nice for me and I spent a good time during the investigation. But I am a ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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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### 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 ...

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### 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
$$ ...

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### 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
...

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### 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 ...