%I #40 Aug 11 2024 22:05:36
%S 1,3,12,56,280,1452,7700,41456,225648,1238420,6840988,37986984,
%T 211842696,1185635388,6655993380,37463920608,211350457824,
%U 1194706644516,6765300359468,38370431711000,217931108199672
%N a(n) = T(n,n-1), array T as in A050143. Also T(2n+1,n), array T as in A055807.
%F From _Vladimir Kruchinin_, Nov 25 2014: (Start)
%F G.f.: x*((-x^2 + 4*x + 1)/(2*sqrt(x^2 - 6*x + 1)) -x/2 + 1/2).
%F For n >= 2, a(n) = C(2*n-3,n-2) + Sum_{i=0..n-2} C(n,i+1)*C(n+i-2,n-2). (End)
%F a(n) ~ (1 + sqrt(2))^(2*n-2) / (2^(1/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 14 2021
%F a(n) = binomial(2*n-3, n-1)*hypergeom([-n+1, -n], [-2*n+3], -1). - _Detlef Meya_, Dec 04 2023
%t a[n_]:=Binomial[2*n-3,n-1]*Hypergeometric2F1[-n+1,-n,-2*n+3,-1];
%t Table[a[n],{n,1,21}] (* _Detlef Meya_, Dec 04 2023 *)
%o (Maxima) a(n):=if n=1 then 1 else sum((binomial(n,i+1))*binomial(n+i-2,n-2),i,0,n-2)+binomial(2*n-3,n-2); /* _Vladimir Kruchinin_, Nov 25 2014 */
%Y Cf. A002003, A006318, A050143, A055807.
%K nonn
%O 1,2
%A _Clark Kimberling_