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    $\begingroup$ Did you have a look in Krattenthaler's "advanced determinant calculus"? arxiv.org/pdf/math/9902004v3.pdf $\endgroup$ Commented Jan 11, 2014 at 9:51
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    $\begingroup$ "Is there an existing proof" implies that the identity is new/unproved. Can you, please, clarify how do you know that the identity holds and give some context? $\endgroup$ Commented Jan 11, 2014 at 15:36
  • $\begingroup$ Many thanks to all for the comments. I have a proof based on an LU decomposition as suggested by Douglas (@Douglas-Zare) with \begin{eqnarray*} L_{ij}=\left\{ \begin{array}{cc} \binom{2i-j-1}{j-1}& \mbox{for } i\le j ,\\ 0 & \mbox{for } i>j, \end{array} \right. \qquad U_{jk}= \left\{ \begin{array}{cc} \frac{k}{j}\binom{j+k-1}{2j-1} &\mbox{for } j \le k,\\ 0 & \mbox{for }j>k. \end{array} \right. \end{eqnarray*} and using induction in k. Gjergji's (@Gjergji-Zaimi) general formula is very nice! $\endgroup$ Commented Jan 17, 2014 at 10:56