Certain diagonal invariants of the iterated polynomial sequence p_n(x) = p_{n-1}(x)^2 - 2 are shown to be weighted Catalan tree sums, equal in closed form to 2(-1)^k/(2k)!.
Journal of Physics A: Mathematical and General , 37(3):657, 2004
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A study of a recursive sequence of polynomials revealing weighted Catalan Numbers
Certain diagonal invariants of the iterated polynomial sequence p_n(x) = p_{n-1}(x)^2 - 2 are shown to be weighted Catalan tree sums, equal in closed form to 2(-1)^k/(2k)!.