All Questions
5
questions
2
votes
0
answers
168
views
Polynomial Division of a "Special" Polynomial
Let $$f(m)=(2n+1)((2n+1)^2-1^2)((2n+1)^2-3^2)\ldots((2n+1)^2-(2m-3)^2)/(2m-1)!$$
for some Positive integers $n,m$
we have to find the coefficients of $t^{1-k}$ quotient on polynomial Division of
$...
- 39
0
votes
1
answer
111
views
Coefficients of $(1+x+x^2)^{2018}$ [closed]
The question is
How many of the coefficients of $(1+x+x^2)^{2018}$ are not divisible by 3?
Somebody asked me the question, and I have no idea how to solve it. I am not sure if the coefficients are ...
- 9,961
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}...
1
vote
1
answer
74
views
Missing notation in one of the test questions
I am looking at STEP (Cambridge produced test for maths) questions and have stumbled upon this question. Does anyone know what notation is missing here in the section i). Please do not provide me with ...
- 3,441
5
votes
3
answers
541
views
Closed form of a sum of binomial coefficients?
I have the following function:
$T_n(d)=\sum\limits_{k=\frac{n-d}{2}}^{\lceil \frac{n}{2} \rceil}{k\choose \frac{n-d}{2}}$
${n \choose 2k}$, where $n,d\in \mathbb{N}^0$, and $n,d$ have the same ...
- 5,292