A302286 a(n) = [x^n] 1/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - ...))))), a continued fraction.
1, 2, 12, 116, 1530, 25422, 507696, 11814728, 313426350, 9324499610, 307171539576, 11091813369276, 435408606414964, 18453269887229478, 839464708754178240, 40786587211854543120, 2107367668847505288726, 115352793604678609311282, 6667002839420189781109800, 405656528458830256952396420
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Keywords
Programs
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Mathematica
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}]
Formula
a(n) = [x^n] (1 - n*x - sqrt(1 - (2*n + 4)*x + n^2*x^2))/(2*x).
a(0) = 1; a(n) = (1/n)*Sum_{k=0..n} (n + 1)^k*binomial(n,k)*binomial(n,k-1).
a(n) = A247507(n,n).
a(n) ~ exp(2*sqrt(n)) * n^(n - 3/4) / (2*sqrt(Pi)). - Vaclav Kotesovec, Jun 08 2019