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Tagged with binomial-coefficients polynomials
188
questions
2
votes
1
answer
462
views
Using the Multinomial Theorem to Calculate a Finite Sum raised to an exponent
I know it's a simple question, but I keep getting different general formulas for the coefficients when I am trying to use the multinomial theorem for the following:
$$
\left(\sum_{k=0}^{M}\frac{(-x^2)^...
- 21
3
votes
2
answers
625
views
Recurrence equation for central trinomial coefficients
I've come across the following exercise:
Give a recurrence equation for the central coefficients $(a_n)$, where for all $n$, $a_n$ is the coefficient of $X^n$ in $(1+X+X^2)^n$.
Here's what I've ...
- 759
15
votes
3
answers
22k
views
Derivation of binomial coefficient in binomial theorem.
How was the binomial coefficient of the binomial theorem derived?
$$\frac{n!}{k!(n-k)!}$$
user26649
8
votes
3
answers
760
views
Intuitive explanation for a polynomial expansion?
Is there an ituitive explanation for the formula:
$$
\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
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
4
votes
3
answers
387
views
Property of a polynomial $f\in\mathbb{Q}[X]$ such that $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$?
We can always view $\binom{x}{k}$ as a polynomial in $x$ of degree $k$. With this in mind, why is it so that a polynomial $f\in\mathbb{Q}[x]$ is such that $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$ ...
- 43
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
5
votes
1
answer
150
views
How to calculate this efficiently?
If in the expansion of $(1 + x)^m \cdot (1 – x)^n $, the coefficients of $ x $ and $ x^2 $are 3 and -6 respectively, then m is ?
I solved it in the following way :
Expanding we get, the coefficient ...
Tretwick Marain