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A323209 a(n) = hypergeometric([-n, n + 1], [-n - 1], n).

%I A323209 #9 Feb 26 2019 03:58:19
%S A323209 1,2,13,190,4641,161376,7312789,409186310,27272680705,2110472708140,
%T A323209 186023930383501,18401769878685172,2018938571514794593,
%U A323209 243319689384354960300,31955654188732155634341,4542582850906442990797126,694922224386422689648830465
%N A323209 a(n) = hypergeometric([-n, n + 1], [-n - 1], n).
%F A323209 a(n) = A323206(n, n).
%F A323209 a(n) = Sum_{j=0..n} (binomial(2*n-j, n) - binomial(2*n-j, n+1))*n^(n-j).
%F A323209 a(n) = Sum_{j=0..n} binomial(n+j, n)*(1 - j/(n + 1))*n^j.
%F A323209 a(n) = 1 + Sum_{j=0..n-1} ((1+j)*binomial(2*n-j, n+1)/(n-j))*n^(n-j).
%F A323209 a(n) = (1/(2*Pi))*Integral_{x=0..4*n} (sqrt(x*(4*n-x))*x^n)/(1+(n-1)*x), n>0.
%F A323209 a(n) ~ (4^(n + 1)*n^(n + 1/2))/(sqrt(Pi)*(1 - 2*n)^2).
%p A323209 # The function ballot is defined in A238762.
%p A323209 a := n -> add(ballot(2*k, 2*n)*n^k, k=0..n):
%p A323209 seq(a(n), n=0..16);
%t A323209 a[n_] := Hypergeometric2F1[-n, n + 1, -n - 1, n];
%t A323209 Table[a[n], {n, 0, 14}]
%Y A323209 Cf. A323206, A238762.
%K A323209 nonn
%O A323209 0,2
%A A323209 _Peter Luschny_, Feb 25 2019