This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A323208 #9 Feb 26 2019 03:58:04
%S A323208 1,5,67,1606,55797,2537781,142648495,9549411950,741894295369,
%T A323208 65620725560578,6511108452179611,716273662860469000,
%U A323208 86527644431076024637,11387523335268377432565,1621766490238904658104583,248507974510512755641561366,40769019250019155227631614225
%N A323208 a(n) = hypergeometric([-n - 1, n + 2], [-n - 2], n).
%F A323208 a(n) = A323206(n, n+1).
%F A323208 a(n) = Sum_{j=0..n+1} (binomial(2*(n+1)-j,n+1)-binomial(2*(n+1)-j,n+2))*n^(n+1-j).
%F A323208 a(n) = Sum_{j=0..n+1} binomial(n+1+j, n+1)*(1 - j/(n+2))*n^j.
%F A323208 a(n) = 1 + Sum_{j=0..n} ((1+j)*binomial(2*(n+1)-j, n+2)/(n+1-j))*n^(n+1-j).
%F A323208 a(n) = (1/(2*Pi))*Integral_{x=0..4*n} (sqrt(x*(4*n-x))*x^(n+1))/(1+(n-1)*x), n>0.
%F A323208 a(n) ~ (4^(n + 2)*n^(n + 3))/(sqrt(Pi)*(1 - 2*n)^2*(n + 1)^(3/2)).
%p A323208 # The function ballot is defined in A238762.
%p A323208 a := n -> add(ballot(2*j, 2*n+2)*n^j, j=0..n+1):
%p A323208 seq(a(n), n=0..16);
%t A323208 a[n_] := Hypergeometric2F1[-n - 1, n + 2, -n - 2, n];
%t A323208 Table[a[n], {n, 0, 16}]
%Y A323208 Cf. A323206, A238762.
%K A323208 nonn
%O A323208 0,2
%A A323208 _Peter Luschny_, Feb 25 2019