All Questions
5
questions
1
vote
2
answers
120
views
Coefficient of polynomial of $x^k$
Consider a polynomial of power n:
$P(x)=1+x+x^2+\dots+x^n$
How do I find coefficient of $x^k$, where $0\le k\le 3n$ of the polynomial $P^3(x)$?
I have tried plugging in different values of $n$ to find ...
- 135
3
votes
1
answer
135
views
Reciprocal binomial coefficient polynomial evaluation
The conventional binomial coefficient can be obtained via
$$
f(x, n) = (1+x)^n = \sum_{i=0}^n { n \choose i} x^i
$$
And the function $f$ can be every efficiently performed on evaluation.
I'm ...
- 1,271
2
votes
1
answer
228
views
Expansion of $(a+b+c+d+e+....)^n$, but with all coefficients equal to 1.
I'm looking for a formula to calculate the sum of $(a+b+c+d+...)^n$ but with coefficients equal to 1.
For example in $(a+b+c)^2$. I want the sum of $a^2 + b^2 + c^2 + ab + bc + ca$. And for $(a+b+c+d)^...
- 123
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
1
vote
3
answers
450
views
Coefficients of polynomial $(x+1)(x+2)...(x+n)$
I was trying to calculate the integral
$$
I(m,n)=\int_0^\infty\frac{x^me^x}{(1-e^x)^n}\mathrm{d}x
$$
It's possible to evaluate this in closed form by using the zeta function, for example:
$$
I(m,4)=m!(...
- 2,148