OFFSET
0,2
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
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 04 2018
STATUS
approved