Editing Catalan number (section)

== Hankel matrix ==

The {{math|''n'' × ''n''}} [[Hankel matrix]] whose {{math|(''i'', ''j'')}} entry is the Catalan number {{math|''C''<sub>''i''+''j''−2</sub>}} has [[determinant]] 1, regardless of the value of {{mvar|n}}. For example, for {{math|1=''n'' = 4}} we have
:<math>\det\begin{bmatrix}1 & 1 & 2 & 5 \\ 1 & 2 & 5 & 14 \\ 2 & 5 & 14 & 42 \\ 5 & 14 & 42 & 132\end{bmatrix} = 1.</math>

Moreover, if the indexing is "shifted" so that the {{math|(''i'', ''j'')}} entry is filled with the Catalan number {{math|''C''<sub>''i''+''j''−1</sub>}} then the determinant is still 1, regardless of the value of {{mvar|n}}.
For example, for {{math|1=''n'' = 4}} we have
:<math>\det\begin{bmatrix}1 & 2 & 5 & 14 \\ 2 & 5 & 14 & 42 \\ 5 & 14 & 42 & 132 \\ 14 & 42 & 132 & 429 \end{bmatrix} = 1.</math>

Taken together, these two conditions uniquely define the Catalan numbers.

Another feature unique to the Catalan–Hankel matrix is that the {{math|''n'' × ''n''}} submatrix starting at {{math|2}} has determinant {{math|''n'' + 1}}.

:<math>\det\begin{bmatrix} 2 \end{bmatrix} = 2</math>

:<math>\det\begin{bmatrix}  2 & 5 \\5 & 14 \end{bmatrix} = 3</math>

:<math>\det\begin{bmatrix}  2 & 5 & 14\\5 & 14 & 42\\ 14 & 42 & 132\end{bmatrix} = 4</math>

:<math>\det\begin{bmatrix}  2 & 5 & 14 & 42 \\ 5 & 14 & 42 & 132 \\ 14 & 42 & 132 & 42 9\\ 42 & 132 & 429 & 1430\end{bmatrix} = 5</math>

et cetera.