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A050147 a(n) = T(n,n-1), array T as in A050143. Also T(2n+1,n), array T as in A055807.

%I A050147 #43 Jul 31 2025 11:53:24
%S A050147 1,3,12,56,280,1452,7700,41456,225648,1238420,6840988,37986984,
%T A050147 211842696,1185635388,6655993380,37463920608,211350457824,
%U A050147 1194706644516,6765300359468,38370431711000,217931108199672
%N A050147 a(n) = T(n,n-1), array T as in A050143. Also T(2n+1,n), array T as in A055807.
%F A050147 From _Vladimir Kruchinin_, Nov 25 2014: (Start)
%F A050147 G.f.: x*((-x^2 + 4*x + 1)/(2*sqrt(x^2 - 6*x + 1)) -x/2 + 1/2).
%F A050147 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 A050147 a(n) ~ (1 + sqrt(2))^(2*n-2) / (2^(1/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 14 2021
%F A050147 a(n) = binomial(2*n-3, n-1)*hypergeom([-n+1, -n], [-2*n+3], -1). - _Detlef Meya_, Dec 04 2023
%F A050147 D-finite with recurrence (-n+1)*a(n) +(2*n+3)*a(n-1) +3*(8*n-29)*a(n-2) +(-10*n+49)*a(n-3) +(n-6)*a(n-4)=0. - _R. J. Mathar_, Jul 31 2025
%t A050147 a[n_]:=Binomial[2*n-3,n-1]*Hypergeometric2F1[-n+1,-n,-2*n+3,-1];
%t A050147 Table[a[n],{n,1,21}] (* _Detlef Meya_, Dec 04 2023 *)
%o A050147 (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 A050147 Cf. A002003, A006318, A050143, A055807.
%K A050147 nonn
%O A050147 1,2
%A A050147 _Clark Kimberling_