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
0
votes
0
answers
40
views
Finding particular solution using domain transformation
$$ φ(n)=5 φ\left(\frac{n}{2}\right)-6 φ\left(\frac{n}{4}\right)+n $$ where $$ \varphi (1) = 2 \\ \text{and} \\ \varphi (2) = 1 $$
With $n=2^x$, I have the following equation. Am I wrong in this ...
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
0
votes
1
answer
77
views
Finding a generating function
Suppose I have 4 numbers, $x_0,x_1,x_2$ and $x_3$, and the sum,
$$x_0+x_1+x_2+x_3$$
I put the constraint that $x_0$ and $x_3$ are either 1 or 0, and $x_0$ and $x_3$ can be equal or between $0$ to $3; ...
- 493
4
votes
0
answers
270
views
How to prove this equality about Eulerian numbers? [duplicate]
I want to prove the following equality where $A(k,m)$ is the Eulerian number :
$$\forall k\ge0,\sum_{k=0}^{\infty}n^k x^k = \frac{\sum_{m=0}^{k-1}A(k,m)x^{m+1}}{(1-x)^{k+1}}$$
I previously proved ...
- 9,427