All Questions
Tagged with binomial-coefficients polynomials
20 questions
0
votes
1
answer
169
views
Partial sums of binomial coefficients and related family of polynomials
Let $a(n)$ be A302117. Here
$$
a(n) = 4(n-1)a(n-1) - \frac{1}{3}\prod\limits_{k=0}^{n-1}(2k-3), \\
a(0) = 0.
$$
Let
$$
T(n,k) = \sum\limits_{i=0}^{k} \binom{n}{i}.
$$
Let $P_n(z)$ be the family of ...
- 34.3k
4
votes
0
answers
168
views
How to prove the following equation (which involves binomials and determinant of 2×2 matrices)?
I have tried many ways to prove the following equation, such as the method of induction and expanding all the terms in the summation,but things got more complicated.I could not find an appropriate ...
- 188k
0
votes
1
answer
98
views
Only special permutations result in a constant expression when permuting coefficients in a sum involving binomials?
Fix $n\geq 1$ and let $p_k(x) := x^k(x-1)^{n-k}$.
Suppose $\pi$ is a permutation on $\{0,1,\dotsc,n\}$, such that
$$
\sum_{k=0}^n (-1)^k \binom{n}{k} p_{\pi(k)}(x) \text{ is a constant}.
$$
Must it be ...
- 13.2k
8
votes
3
answers
642
views
Combinatorial identity with connection coefficients and falling factorial $\langle i x\rangle_n$
Let $j, k ,n$ be nonnegative integers such that $0 \leq j, k \leq n \leq k +j $. Pick integer $m$ such that $0 \leq m \leq k + j - n$.
Let $\langle x \rangle_m$ denote the falling factorial $x(x-1)\...
- 10.5k
5
votes
0
answers
183
views
On the polynomials $\sum_{k=0}^n\binom{n+k}k^m q^k$
A sequence of polynomials
$$P_0(q),\ P_1(q),\ P_2(q),\ \ldots$$
with real coefficients is called $q$-log-convex if for each $n=1,2,3,\ldots$ every coefficient of the polynomial $P_{n+1}(q)P_{n-1}(q)-...
- 15.6k
6
votes
1
answer
449
views
A summation involving fraction of binomial coefficients
I need to prove the following statement.
Let $ n, g, m, a ,t$ be integers. Prove that the following statement is true for all $ n \geq g(1+2m)+1 $, $ g\geq 2t $, $ m\geq t $, $ 0\leq a <t $, and $ ...
- 34.3k
4
votes
0
answers
134
views
Irreducibility of polynomials associated to binomial coefficients
Let $n \geq 2$.
Let $M_n$ be the $(n+1) \times (n+1)$ matrix with entries $\binom{l}{k}$ for $0 \leq l,k \leq n$ and $U_n=M_n + M_n^T$ and let $f_n(x)$ denote the characteristic polynomial of $U_n$.
...
- 26.5k
1
vote
0
answers
183
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polynomial approximation of hypergeometric function 2F1
I have the following function $T(k_1,k_2)$ resulting from multiphoton transition matrix elements calculations:
$T(k_1,k_2)=\gamma^{-k_2}\sum_{j=0}^{k_1}(j+2)_{l+1}\binom{k_1}{j}(k_1+1)_3(\gamma-1)^{j}{...
- 11
0
votes
1
answer
167
views
Restrictions on exponents in multinomial formula
From the multinomial formula we have
$$(x_1 + x_2 + \dotsb + x_m)^n
= \sum_{k_1+k_2+\dotsb+k_m=n, \ k_1, k_2, \dotsc, k_m \geq 0} {n \choose k_1, k_2, \dotsc, k_m}
\prod_{t=1}^m x_t^{k_t}\,.$$
I ...
- 34.3k
3
votes
0
answers
144
views
Flat polynomials with factors of big height
Let $p(x)$ be a polynomial of degree $n$ with all coefficients in $\{-1,0,1\}$ (such polynomials are sometimes called flat). I am wondering how big the coefficients of a factor of $p$ can be. Call ...
- 13.4k
3
votes
3
answers
397
views
Chebyshev polynomials and ballot numbers
I have asked this question a short time ago on mathstackexchange, but it has already fallen into the abyss of answered and uncommented questions. So I take the risk to ask it on mathoverflow.
Playing ...
- 17k
4
votes
2
answers
470
views
Closed form for $\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$
As stated, I wonder if there is a closed form for the generating function $F_{\alpha,\beta}(x):=\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$ where $\alpha,\beta \in\mathbb{N}$. Calling ...
0
votes
0
answers
53
views
Interlaced binomial expansion
For $a_1,a_2,\ldots,a_n < 1$ let
$$f(a_1,a_2,\ldots,a_n) = \sum_{k_1,k_2,...,k_n = 1}^\infty \frac{(k_1 + k_2)!}{k_1! k_2!} \frac{(k_2 + k_3)!}{k_2! k_3!}\ldots \frac{(k_{n-1} + k_{n})!}{k_{n-1}! ...
- 503
21
votes
2
answers
2k
views
Real rootedness of a polynomial
Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by:
$$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$
I've found with Sage that for every $...
- 85.6k
0
votes
1
answer
296
views
Showing equality of Eberlein polynomials
I have thought about the following question a long time and still got no progress.
Currently I am writing my master thesis about association schemes in combinatorics and need an equality which seems ...
18
votes
3
answers
860
views
$\prod_k(x\pm k)$ in binomial basis?
Let $x$ be an indeterminate and $n$ a non-negative integer.
Question. The following seems to be true. Is it?
$$x\prod_{k=1}^n(k^2-x^2)=\frac1{4^n}\sum_{m=0}^n\binom{n-x}m\binom{n+x}{n-m}(x+2m-n)^...
- 3,255
8
votes
0
answers
597
views
Does anyone know this sequence of polynomials?
A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$.
\begin{align}
P_{0,0}&=1\\
\text{for $n\geq 1$}...
- 19.3k
1
vote
1
answer
525
views
Polynomial convex coefficients
Assume we have an arbitrary high order polynomial $$f(L)=1-L\theta_1-L^2\theta_2-L^3\theta_3-...-L^N\theta_N$$ and we know all roots of this polynomial site outside the unit circle. It is obvious that ...
- 25.8k
47
votes
1
answer
4k
views
How to prove this polynomial always has integer values at all integers?
Let $m$ be any positive integer.
$$
P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}.
$$
Question: $P_m(x)$ always ...
CommunityBot
- 1
2
votes
3
answers
634
views
Variant of binomial coefficients
I've recently come across a variant of the binomial polynomials, and I'm curious if anyone has seen these before. If so, I'd love a reference, a name, etc.
First recall the following. If z is a ...
- 2,955