All Questions
Tagged with hankel-matrices co.combinatorics
15 questions
2
votes
0
answers
241
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Determinants of band matrices which are related to Hankel matrices of Catalan numbers
Let $A_{n,m}$ be the band matrix $$ A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \right)_{0\leq {i,j} \leq {n-1}}.$$
For example,
$$A_{6,2}=\left ( \begin{matrix} 2 & 3 & 1& 0 &...
- 5,612
1
vote
0
answers
78
views
Shifted Hankel determinants for convolutions of Catalan numbers
It is well known that for $m\in \mathbb N$ the Hankel determinants $$D_m(n)= \det\left(C_{i+j+m}\right)_{0\leq i,j\leq {n-1}}$$ satisfy $D_m(n)=p_m(n)$, where $p_m(n)=\prod_{1 \leq i \leq j \leq {m-1}}...
- 5,612
7
votes
0
answers
252
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Hankel determinants for some convolutions of Catalan numbers
Let $c(x)=\frac{1-\sqrt{1-4x}}{2x}$ be the generating function of the Catalan numbers and let $$x^k c(x)^{2k}=(c(x)-1)^k =\sum_{n\geq0}c(k,n)x^n.$$
Consider the determinants $$D(k,n,m)= \det\left(c(k,...
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1
vote
1
answer
185
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A continued J fraction for $a_n = \frac{1}{(n+1)^2}$?
The following is called a J continued fraction:
$$\cfrac{\alpha_0}{1+a_0x-\cfrac{b_1x^2}{1+a_1x-\cfrac{b_2x^2}{1+a_2x-\cdots}}}$$
where the constants are real numbers. Let $\alpha_n= \frac{1}{(n+1)^2}$...
- 233
5
votes
1
answer
408
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An interesting Hankel determinant
Let $h(n,t) = \sum\limits_{j = 0}^n {\binom
{\lfloor {\frac{n}{2}} \rfloor }{j}\binom
{\lfloor {\frac{n+1}{2}}\rfloor }{j}t^j \\ }.$
I am interested in the Hankel determinants $${D_k}(n,t) = \det \...
- 5,612
3
votes
2
answers
370
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Number of bounded Dyck paths with negative length as Hankel determinants
This is a continuation of my post Number of bounded Dyck paths with "negative length".
Let $C_{n}^{(2k+1)}$ be the number of Dyck paths of semilength $n$ bounded by $2k+1.$ They satisfy a ...
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12
votes
1
answer
553
views
A matrix identity related to Catalan numbers
Let $$C_n=\frac{1}{2n+1}\binom{2n+1}{n}$$ be a Catalan number. It is well-known that $$(\sum_{n\ge{0}}C_n x^n)^k=\sum_{n\ge{0}}C(n,k)x^n$$ with $$C(n,k)=\frac{k}{2n+k}\binom{2n+k}{n}.$$
It is also ...
- 5,612
13
votes
1
answer
385
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Some more binomial coefficient determinants
The setup is similar to this question, but generalizes the size of the Hankel matrix. We'll define
$$d(n,k,r):=\det\left(\binom{2i+2j+k+r}{i+j}\right)_{i,j=0}^{kn-1}.$$
Edit: Thanks to Johann ...
- 13.4k
3
votes
1
answer
187
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The Golay-Rudin-Shapiro sequence as “Hankel transform”
Let ${\left( {{a_n}} \right)_{n \geqslant 0}}$ be a sequence of real numbers and ${H_n} = \det \left( {{a_{i + j}}} \right)_{i,j = 0}^{n - 1}$ the $n-$th Hankel determinant. The sequence ${\left( {{...
- 5,612
9
votes
0
answers
213
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Some quotients of Hankel determinants
This question has been inspired by Hankel determinants of binomial coefficients.
For a sequence $\{h_{n}\}_{n=0}^{\infty}$ denote by $H_n$ the Hankel matrix
$$H_{n}:=\begin{pmatrix}
h_{0} & h_{...
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4
votes
1
answer
247
views
Hankel determinant evaluation of special lattice paths
Let $n$ be a positive integers and $T=T_{n,n}$ be the $n\times n$ table in the first quadrant composed of $n^2$ unit squares, whose $(x,y)$-blank is locate in the $x^{th}$-column from ...
- 181
5
votes
2
answers
635
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Some curious Hankel determinants
Let $f(n,q)=\prod_{j=1}^na(q^j)$ for a polynomial $a(q)$ and let $d(n)=\det(f(i+j,q))_{i,j=0}^n$ be its Hankel determinant.
Computer experiments suggest that
$$\lim_{q\to1}\frac{d(n)}{(q-1)^\binom{n+...
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11
votes
1
answer
330
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a Hankel matrix of involution numbers
Let $I_k$ denote the enumeration of involutions among permutations in $\mathfrak{S}_k$. I always enjoy these numbers. Of course, here is yet another cute experimental finding for which I ask validity. ...
- 43.2k
12
votes
2
answers
779
views
Determinant of a checkerboard Hankel matrix with Catalan numbers
My goal is to compute
\begin{equation}
I = \det \left(\mathbf{I} + \mathbf{A}\right)
\end{equation}
where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers:
$$
\left\{
...
- 840
5
votes
0
answers
482
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A class of determinants associated to Catalan-like Hankel determinants
The following matrices are related to some Catalan-like Hankel matrices. My question is whether direct computations of determinants of such matrices (i.e. without recourse to Hankel determinants) ...
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