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1 answer
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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 ...
Notamathematician's user avatar
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 ...
tongjun's user avatar
  • 41
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 ...
Per Alexandersson's user avatar
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)-...
Zhi-Wei Sun's user avatar
  • 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 $ ...
Arda Aydin's user avatar
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 $U_n$. ...
Mare's user avatar
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1 vote
0 answers
183 views

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}{...
Omer Amit's user avatar
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 ...
eyejay's user avatar
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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 ...
Wolfgang's user avatar
  • 13.4k
3 votes
3 answers
398 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 ...
Libli's user avatar
  • 7,300
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 ...
Milo Moses's user avatar
  • 2,902
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}! ...
tomate's user avatar
  • 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 $...
Luis Ferroni's user avatar
  • 1,889
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 ...
McRatchet's user avatar
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)^...
T. Amdeberhan's user avatar
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)\...
Nick R's user avatar
  • 1,187
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 ...
Chitsai Liu's user avatar
  • 2,183
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 ...
TPArrow's user avatar
  • 161
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 $n\geq 1$}...
John Pardon's user avatar
  • 18.7k
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 ...
Daniel Erman's user avatar
  • 2,955