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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) can be found in the literature (except the simple cases p=0 or p=1 and k=0).

Let $H_n^{(p)} (k,c)$ be defined by $H_n^{(0)} (k,c) = \left( {h(k,i,j)} \right)_{i,j = 0}^{n - 1} $ where $h(k,i,j) = 1$ if $i + j = k + 2l$ for a nonnegative integer $l$ and $\left| {j - i} \right| \le k$, and $h(k,i,j) = 0$ else.

For $p > 0$ let $H_n^{(p)} (0,c) = cH_n^{(p - 1)} (0,c) + H_n^{(p - 1)} (1,c)$ and

$H_n^{(p)} (k,c) = H_n^{(p - 1)} (k - 1,c) + cH_n^{(p - 1)} (k,c) + H_n^{(p - 1)} (k + 1,c)$.

To indicate the connection with Hankel determinants consider e.g. the sequence (a(n))=(1,0,1,0,2,0,5,0,14,0,…) of Catalan numbers. Then $\det \left( {a(i + j + p)} \right)_{i,j = 0}^{n - 1} = \det H_n^{(p)} (0,0).$

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Have you tried looking for in Christian Krattenthaler's surveys of determinants (and also in his arts on Catalan's numbers)? –  Wadim Zudilin Apr 24 '10 at 11:20
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