All Questions
Tagged with binomial-coefficients polynomials
188
questions
3
votes
5
answers
213
views
Coefficient of $x^{21}$ in $(1+x+x^2+\dots+x^{10})^4$
Find the coefficient of $x^{21}$ in $(1+x+x^2+\dots+x^{10})^4$
I tried splitting the terms inside the bracket into two parts $1+x+\dots+x^9$ and $x^{10}$, and then tried binomial theorem, but that ...
- 545
2
votes
1
answer
108
views
Find all values of $n$ where $2n^2+7n+3$ is prime. [closed]
Problem: Find all values of $n$ where $2n^2+7n+3$ is prime.
What I know:
All prime numbers are only divided by one and themselves.
To reach a solution, I need to factor the polynomial and equal the ...
- 39
2
votes
0
answers
77
views
Closed expression for a combinatorial sum
The following equality is true for every positive integer $n$ :
$$\sum_{k=0}^n {n \choose k} = 2^n $$
It is a special case ($p = 0$) of the sequence :
$$S_{p, n}=\sum_{k=0}^n k^p {n \choose k} $$
For ...
- 389
2
votes
1
answer
342
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Polynomial representation of $\sum\limits_{n=m+1}^N \binom {n-1} m$
Background: I am looking for a polynomial representation of
$$
\sum_{n=m+1}^N \binom {n-1} m \tag{1}
$$
where $m\in\mathbb{Z^+}$ is a positive integer $2,3,4,\ldots $ that is greater than or equal to $...
- 10.1k
1
vote
2
answers
119
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
114
views
A fun double sum involving binomial coefficients
I came across the following double sum expression:
$$
S(k) := \sum_{l=0}^{2(k+1)-1} \sum_{i=0}^{l} (-1)^i\binom{2(k+1)-i}{2(k+1)-l}
\left[ 3\binom{2k}{i-2}+ 3k\left(\binom{2k}{i-1} -
\binom{2k-1}{i-2}\...
- 336
2
votes
4
answers
252
views
Coefficient of $x^k$ in polynomial
Let $k, n, m \in \mathbb{N}, k \le n.$ Find the formula for coefficient of $x^k$ in $(x^n + x^{(n-1)} + ... + x^2 + x + 1)^m$.
answer is in this question: faster-way-to-find-coefficient-of-xn-in-1-x-...
- 311
2
votes
2
answers
108
views
A general formula for $\mathcal{F}_{n} = \prod_{i=1}^n (a_ix + b_iy)$
So I am trying to simplify following product, $$\mathcal{F}_{n} = \prod_{i=1}^n \left(a_ix + b_iy\right)$$ in terms of products and summation. This is what I have come up with so far.
We see that for ...
- 141
2
votes
1
answer
95
views
General formula for $\sum_{k=0}^n k^a \binom{n}{k}$ is somehow hypergeometric? [duplicate]
Original question is a duplicate of this question. Please refer to after the edit
Original question:
I was investigating formulas of the form
$\sum_{k=0}^n k^a \binom{n}{k}$ after noticing on ...
- 245
3
votes
1
answer
75
views
How to prove that this binomial sum remains positive for $x>1$?
Let's say you have this function for real numbers $x>1$, for some positive integer $n \geq 1$
$$
\sum_{k=0}^{\left \lfloor n/2 \right \rfloor} {x \choose 2k+\frac{1-(-1)^n}{2}}
$$
How would you ...
- 633
0
votes
1
answer
87
views
Simplifying a sum of binomials, is there a closed form?
I am struggling with a formula that I derived, which (I believe) can be simplified further.
In principle, I want to determine the coefficients of the following polynomial:
\begin{align}
p(x) = (1+x+.....
- 31
2
votes
1
answer
121
views
Alternative expressions for Krawtchouk (Kravchuk) polynomials
For fixed non-negative integers $n$ and $q \geq 2$, the $k$-th Krawtchouk (Kravchuk) polynomial is defined as
$$K_k = \sum_{j=0}^k (-1)^j (q-1)^{k-j} \binom{X}{j} \binom{n-X}{k-j} \in \mathbb{Q}[X]$$
...
- 23k
1
vote
0
answers
67
views
Closed formula for $\sum_{k=0}^{\lfloor n/2\rfloor} x^k\binom{n-k}{k}$
Let $$P_n(x) = \sum_{k=0}^{\lfloor n/2\rfloor} x^k\binom{n-k}{k}.$$ It's known that $P_n(1) = F_{n+1}$, the $(n+1)$th Fibonacci number, see for example here. Can we find a closed form for this ...
- 10.5k
1
vote
1
answer
43
views
Prove that the 5-adic valuation of $(10^{k+t}+10^t+1)^c-1$ is $t+v_5(c)$
Here is a result of mine that I really wish to be strictly proven (I need it as an intermediate lemma for a theorem).
Let $v_5(a)$ indicate the $5$-adic valuation of $a \in \mathbb{Z}^+$.
How can we ...
- 1,160