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MATHEMATICA
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Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-n x, 1 - x, {i, 1, n}]), {x, 0, n}], {n, 0, 17}]
Table[SeriesCoefficient[2/(1 + x + Sqrt[1 - x (2 + 4 n - x)]), {x, 0, n}], {n, 0, 17}]
Table[Sum[(-1)^(n - k) (n + 1)^k Binomial[n, k] Binomial[n + k, k]/(k + 1), {k, 0, n}], {n, 0, 17}]
Table[(-1)^n Hypergeometric2F1[-n, n + 1, 2, n + 1], {n, 0, 17}]
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