Recall that the $n$-th Catalan number $C_n=\frac{1}{n+1}{2n\choose n}$ counts the number of paths connecting $(0, 0)$ to $(n, n)$ that travel along the grid of integer lattice points of $R^2$ where each path moves up or right in one-unit steps and no path extends above the line $y = x$.

In linear algebra, a Hankel matrix of Catalan numbers is defined as following: $$H_n^t=(C_{i+j+t})_{0\leq i,j\leq n-1}= \begin{bmatrix} c_{t} & c_{t+1} & c_{t+2} & \dots & c_{t+n-1} \\ c_{t+1} & c_{t+2} & c_{t+3} & \dots & c_{t+n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ c_{t+n-1} & c_{t+n} & c_{t+n+1} & \dots & c_{t+2n-2} \end{bmatrix} $$

How can I calculate the Hankel determinant of Catalan numbers for $t=1$?

Is it possible obtain the Hankel determinant of Catalan numbers for $t>1$?

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    $\begingroup$ Maybe Lindstrom's lemma might help: en.wikipedia.org/wiki/… $\endgroup$ – Lord Shark the Unknown Oct 12 '17 at 21:28
  • $\begingroup$ The case $n=2$ gives rise to this sequence oeis.org/A005700 $\endgroup$ – Donald Splutterwit Oct 12 '17 at 21:33
  • $\begingroup$ I have studied Lindstrom-Gessel-Viennot Lemma, but I did not understand how to prove it!! Is there any way to proving? $\endgroup$ – d.y Oct 12 '17 at 21:35
  • $\begingroup$ \begin{eqnarray*} H_{t,n} = \left( \frac{t(t+1)}{2} \right) ! \prod_{i=1}^{t} \frac{(2(n+i-1))!}{(n+2i-2)!(n+2i-1)!}. \end{eqnarray*} I really don't know how to derive this result, but can offer plenty of a circumstantial evidence to support the claim. $\endgroup$ – Donald Splutterwit Oct 13 '17 at 8:44

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