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A323217 a(n) = hypergeometric([-n, n + 1], [-n - 1], n + 1).

%I A323217 #14 Feb 26 2019 03:58:08
%S A323217 1,3,25,413,10746,387607,17981769,1022586105,68964092542,
%T A323217 5384626548491,477951767068986,47546350648784341,5240644323742274500,
%U A323217 634033030117301108127,83540992651137240168361,11908866726507685451458545
%N A323217 a(n) = hypergeometric([-n, n + 1], [-n - 1], n + 1).
%F A323217 a(n) = A323206(n+1, n).
%F A323217 a(n) = Sum_{j=0..n} (binomial(2*n-j, n) - binomial(2*n-j, n+1))*(n+1)^(n-j).
%F A323217 a(n) = Sum_{j=0..n} binomial(n+j, n)*(1 - j/(n + 1))*(n + 1)^j.
%F A323217 a(n) = 1 + Sum_{j=0..n-1} ((1+j)*binomial(2*n-j, n+1)/(n-j))*(n+1)^(n-j).
%F A323217 a(n) = (1/(2*Pi))*Integral_{x=0..4*(n+1)} (sqrt(x*(4*(n+1)-x))*x^n)/(1+n*x).
%F A323217 a(n) ~ (4^(n+1)*(n+1)^(n+2))/(sqrt(Pi)*(2*n+1)^2*n^(3/2)).
%p A323217 # The function ballot is defined in A238762.
%p A323217 a := n -> add(ballot(2*j, 2*n)*(n+1)^j, j=0..n):
%p A323217 seq(a(n), n=0..16);
%t A323217 a[n_] := Hypergeometric2F1[-n, n + 1, -n - 1, n + 1];
%t A323217 Table[a[n], {n, 0, 16}]
%Y A323217 Cf. A323206, A238762.
%K A323217 nonn
%O A323217 0,2
%A A323217 _Peter Luschny_, Feb 25 2019