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Hankel matrix

The n × n Hankel matrix whose (i, j) entry is the Catalan number C has determinant 1, regardless of the value of n. For example, for n = 4 we have

Moreover, if the indexing is "shifted" so that the (i, j) entry is filled with the Catalan number C then the determinant is still 1, regardless of the value of n.For example, for n = 4 we have

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

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

et cetera.