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MATHEMATICA
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Table[SeriesCoefficient[1/(1 - n x + ContinuedFractionK[-x, 1 - n x, {k, 1, n}]), {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[(1 - n x - Sqrt[1 - (2 n + 4) x + n^2 x^2])/(2 x), {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[(1/n) Sum[(n + 1)^k Binomial[n, k] Binomial[n, k - 1], {k, 0, n}], {n, 1, 19}]]
Table[(n + 1) Hypergeometric2F1[1 - n, -n, 2, n + 1], {n, 0, 19}]
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