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prove that following numbers are equal: (unordered) pairs of lattice paths with n+1 steps each, starting at (0,0), using steps (0,1) or (1,0), ending at the same point and only intersecting at the beginning and end

and same pairs of paths with n-1 steps (starting at (0,0), using steps (0,1), (1,0) each such that one path never rises above the other path.

is it possible to write it as $c_n = \sum\limits_{i = 0}^{n-1} c_{n-1-i}c_i$ (where, for example, first factor ($c_{n-1-i}$) represents number of steps on the x-line and second factor is the number of steps on the y-line)?

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