All Questions
19
questions
2
votes
0
answers
77
views
Closed expression for a combinatorial sum
The following equality is true for every positive integer $n$ :
$$\sum_{k=0}^n {n \choose k} = 2^n $$
It is a special case ($p = 0$) of the sequence :
$$S_{p, n}=\sum_{k=0}^n k^p {n \choose k} $$
For ...
- 389
2
votes
1
answer
342
views
Polynomial representation of $\sum\limits_{n=m+1}^N \binom {n-1} m$
Background: I am looking for a polynomial representation of
$$
\sum_{n=m+1}^N \binom {n-1} m \tag{1}
$$
where $m\in\mathbb{Z^+}$ is a positive integer $2,3,4,\ldots $ that is greater than or equal to $...
- 10.1k
1
vote
1
answer
114
views
A fun double sum involving binomial coefficients
I came across the following double sum expression:
$$
S(k) := \sum_{l=0}^{2(k+1)-1} \sum_{i=0}^{l} (-1)^i\binom{2(k+1)-i}{2(k+1)-l}
\left[ 3\binom{2k}{i-2}+ 3k\left(\binom{2k}{i-1} -
\binom{2k-1}{i-2}\...
- 336
3
votes
1
answer
75
views
How to prove that this binomial sum remains positive for $x>1$?
Let's say you have this function for real numbers $x>1$, for some positive integer $n \geq 1$
$$
\sum_{k=0}^{\left \lfloor n/2 \right \rfloor} {x \choose 2k+\frac{1-(-1)^n}{2}}
$$
How would you ...
- 633
1
vote
1
answer
89
views
Asymtotic of some binomial sum
Assume $n$ is a positive odd integer, I need to find the asymptotic as $n$ goes to infinity of the sum
$$s(n,x)=\frac1x\sum_{k=0}^n (-1)^k\binom{-x-\frac12}{k}\binom{x-\frac12}{n-k},$$
where the ...
- 134
5
votes
2
answers
169
views
Prove that $\sum_{k=1}^n\frac{\prod_{1\leq r\leq n, r\neq m}(x+k-r)}{\prod_{1\leq r\leq n, r\neq k}(k-r)}=1$
For arbitrary $x$ and $1\leqslant m\leqslant n$, prove the following:
$$\sum_{k=1}^n\frac{\prod_{1\leq r\leq n, r\neq m}(x+k-r)}{\prod_{1\leq r\leq n, r\neq k}(k-r)}=1$$
I'm looking for a proof that ...
- 848
1
vote
1
answer
97
views
Sum with binomial coefficient using identity
I want to prove:
$\displaystyle \sum_{k=0}^n (-1)^k \binom{x}{k} = (-1)^n \binom{x-1}{n}$
using: $(1-z)^x \cdot \frac{1}{1-z} = (1-z)^{x-1}$
I know how to do it with induction but i somehow can't ...
- 121
1
vote
1
answer
33
views
Simplify $\sum_{t=k}^{n} (\binom{n}{t} \cdot a^{t-1} \cdot (1 - a)^{n - t - 1} \cdot (t - n \cdot a))$
I was working on my probability theory homework and I found probability density function that looks as following
$$\sum_{t=k}^{n} \binom{n}{t} \cdot a^{t-1} \cdot (1 - a)^{n - t - 1} \cdot (t - n \...
- 983
6
votes
2
answers
121
views
Is the sum $\sum_{k=0}^n{n \choose k}\frac{(-1)^{n-k}}{n+k+1}$ always the reciprocal of an integer $\big(\frac{(2n+1)!}{(n!)^2}\big)$?
Denote the sum $$S_n := \sum_{k=0}^n{n \choose k}\frac{(-1)^{n-k}}{n+k+1}$$
This value arose in some calculations of polynomial coefficients. I'm not used to dealing with expressions of this sort. ...
- 1,044
6
votes
4
answers
258
views
If $P(x)$ is any polynomial of degree less than $n$, show that $\sum_{j=0}^n (-1)^j\binom{n}{j}P(j)=0$. [duplicate]
If $P(x)$ is any polynomial of degree less than $n$, then prove that
$$\sum_{j=0}^n (-1)^j\binom{n}{j}P(j)=0$$
My approach was to try and prove this separately for $j^k\ \ \forall\ \ k<n$, instead ...
- 641
0
votes
2
answers
51
views
The Polynomials $P_n(x+y)$
Let $$\displaystyle P_n(x) = \sum_{k=0}^n \binom{n}{k}x^k.$$
We need to show that $$P_n(x+y) = \sum_{k=0}^n\binom{n}{k}P_k(x)y^{n-k}.$$
In the proof, we have $$\begin{array}{rcl}
P_n(x+y) &=&...
- 11
32
votes
4
answers
2k
views
Questions on a self-made theorem about polynomials
I recently came up with this theorem:
For any complex polynomial $P$ degree $n$:
$$ \sum\limits_{k=0}^{n+1}(-1)^k\binom{n+1}{k}P(a+kb) = 0\quad \forall a,b \in\mathbb{C}$$
Basically, if $P$ is ...
user422749
3
votes
0
answers
108
views
Is there a simple formula for this polynomial sum?
The following seemingly arbitrary polynomial has popped up in my research on period polynomials of modular forms, and I would like to try to understand it a little better. I hope it is OK to ask about ...
- 3,559
2
votes
2
answers
119
views
An Identity for a Fibbonacci-Type Polynomial
Problem:
The polynomials $p_{n}\left(x\right)$ are defined recursively by the linear homogenous order 2 difference equation $$p_{n+1}\left(x\right)=2\left(1-2x\right)p_{n}\left(x\right)-p_{n-1}\left(...
- 124
2
votes
1
answer
116
views
Coefficients of a polynomial
Consider the polynomial $$ P_n(t)=(-1)^n (2t-1) \frac{(2t-1)^{n-1}-t^{2n-2}}{(t-1)^2},$$
that is up to a factor $(-t^2)^{n-1}$ the Tutte polynomial $\frac{x^n-x}{x-1}$ of the cyclic graph evaluated in ...
- 113