All Questions
Tagged with binomial-coefficients polynomials
188
questions
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
15
votes
3
answers
22k
views
Derivation of binomial coefficient in binomial theorem.
How was the binomial coefficient of the binomial theorem derived?
$$\frac{n!}{k!(n-k)!}$$
user26649
15
votes
3
answers
2k
views
Polynomial in $\mathbb{Q}[x]$ sending integers to integers?
We can view the binomial coefficient $\binom{x}{k}$ has a polynomial in $x$ with degree $k$. So taking some $f\in\mathbb{Q}[x]$, why is $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$, precisely when the ...
- 1,097
15
votes
5
answers
4k
views
Inverse of the Pascal Matrix
Let $P_n$ be the $(n+1) \times (n+1)$ matrix that contains the numbers of Pascal's triangle in the upper triangle. For example in the case of $n=3$
$$
P_3 =
\begin{pmatrix}
1 & 1 & 1 & 1 \...
- 3,123
15
votes
4
answers
790
views
Find coefficient of $x^{20}$
Find the coefficient of $x^{70}$ in the expansion
$$(x-1)(x^2-2)(x^3-3)(x^4-4)\cdots (x^{12}-12)$$
$\mathcal {\text {Now I have solved this question}}$. What I did was I noticed that the highest ...
- 9,803
12
votes
0
answers
393
views
How to prove this $p^{j-\left\lfloor\frac{k}{p}\right\rfloor}\mid c_{j}$
Let $p$ be a prime number and $g\in \mathbb{Z}[x]$.
Let $$\binom{x}{k}=\dfrac{x(x-1)(x-2)\cdots(x-k+1)}{k!} \in \mathbb{Q}[x]$$ for every $k \geq 0$.
Fix an integer $k$. Write the integer-valued ...
- 459
11
votes
4
answers
12k
views
Finding coefficient of polynomial?
The coefficient of $x^{12}$ in $(x^3 + x^4 + x^5 + x^6 + …)^3$ is_______?
My Try:
Somewhere it explain as:
The expression can be re-written as: $(x^3 (1+ x + x^2 + x^3 + …))^3=x^9(1+(x+x^2+x^3))^3$
...
- 4,785
11
votes
3
answers
827
views
A Curious Binomial Sum Identity without Calculus of Finite Differences
Let $f$ be a polynomial of degree $m$ in $t$. The following curious identity holds for $n \geq m$,
\begin{align}
\binom{t}{n+1} \sum_{j = 0}^{n} (-1)^{j} \binom{n}{j} \frac{f(j)}{t - j} = (-1)^{n} \...
- 17.1k
11
votes
3
answers
1k
views
Smoothstep sigmoid-like function: Can anyone prove this relation?
The Smoothstep sigmoid-like function is defined as the polynomial
$$ \begin{align}
\operatorname{S}_N(x) &= x^{N+1} \sum_{n=0}^{N} \binom{N+n}{n} \binom{2N+1}{N-n} (-x)^{n} \qquad N \in \mathbb{Z}...
11
votes
1
answer
202
views
Is the given binomial sum almost everywhere negative as $K\to\infty$?
The binomial sum is as follows:
$$\mathcal {L}^K(\theta)= \sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i\left((1-\theta)^{K-i}-\frac{1}{2}(1-\theta)^{-K}(1-2\theta)^{K-i}\right)$$
which can also ...
- 7,892
10
votes
1
answer
1k
views
Vandermonde identity in a ring
Let $R$ be a commutative $\mathbb{Q}$-algebra. For $r \in R$ and $n \in \mathbb{N}$ we can define the binomial coefficient $\binom{r}{n}$ as usual by $\binom{r}{0}=1$ and $\binom{r}{n+1}=\frac{r-n}{n+...
- 165k
10
votes
0
answers
331
views
Expanding a product of linear combinations with coefficients $1$ and $-1$
For any odd natural number $n$, denote $t \equiv \frac{n-1}{2}$. Let $K$ be a field such that $\operatorname{char} K \neq 2$. Working over the polynomial ring $K\left[x_1,x_2,...,x_{n} \right]$, ...
- 124
9
votes
3
answers
415
views
An interesting problem of polynomials
In the polynomial
$$
(x-1)(x^2-2)(x^3-3) \ldots (x^{11}-11)
$$
what is the coefficient of $x^{60}$?
I've been trying to solve this question since a long time but I couldn't. I don't know whether ...
- 1,135
9
votes
3
answers
646
views
sum of rational terms in $\left(\sqrt{2}+\sqrt{27}+\sqrt{180}\right)^{10}$
Find a sum of the rational terms in the following expression after full expanding. $$\left(\sqrt{2}+\sqrt{27}+\sqrt{180}\right)^{10}$$
Since the term should be rational, each power should be even.
...
- 13.2k
8
votes
3
answers
760
views
Intuitive explanation for a polynomial expansion?
Is there an ituitive explanation for the formula:
$$
\frac{1}{\left(1-x\right)^{k+1}}=\sum_{n=0}^{\infty}\left(\begin{array}{c}
n+k\\
n
\end{array}\right)x^{n}
$$
?
Taylor expansion around x=0
...
- 803