The binomial-coefficients tag has no wiki summary.

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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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103 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 & = & ...

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331 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 $0 \leq x \leq k-1$?

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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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244 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 ...

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158 views

### 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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342 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 ...

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**1**answer

120 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 ...

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306 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 ...

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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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**1**answer

225 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 ...

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530 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 ...

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164 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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**1**answer

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### 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,
...

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802 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 ...

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### 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 ...

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### 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} ...

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434 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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
$
...

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**1**answer

258 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 ...

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239 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}} = ...

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### 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.

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**3**answers

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### 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 ...

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### 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} ...

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277 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 ...

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**2**answers

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### 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 ...

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### 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 ...

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### 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 ...