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A270447 Binomial transform(2) of Catalan numbers.

%I A270447 #30 May 30 2022 05:30:43
%S A270447 1,3,11,43,174,721,3044,13059,56837,250690,1119612,5059561,23119628,
%T A270447 106753404,497762380,2342096579,11113027686,53138757319,255892224332,
%U A270447 1240217043450,6046131132030,29631889507380,145923474439800,721733515299225,3583733352377724
%N A270447 Binomial transform(2) of Catalan numbers.
%H A270447 G. C. Greubel, <a href="/A270447/b270447.txt">Table of n, a(n) for n = 0..1000</a>
%F A270447 a(n) = Sum_{k=0..n} (T(n,k)*C(k)), where C(k) is Catalan numbers (A000108), T(n,k) - triangle of A092392.
%F A270447 a(n) = Sum_{k=0..n} ((binomial(2*k,k)/(k+1)*binomial(2*n-k,n))).
%F A270447 G.f.: C(C(x))*(1-C(x))^2/(((1-C(x))^2)-x)/x, where C(x)=(1-sqrt(1-4*x))/2.
%F A270447 Recurrence: 3*(n-1)*n*(n+1)*(2*n - 3)*a(n) = 16*(n-1)*n*(5*n^2 - 10*n + 3)*a(n-1) - 16*(n-1)*(2*n - 1)*(11*n^2 - 33*n + 24)*a(n-2) + 8*(2*n - 3)*(2*n - 1)*(4*n - 9)*(4*n - 7)*a(n-3). - _Vaclav Kotesovec_, Mar 17 2016
%F A270447 a(n) ~ 2^(4*n + 1/2) / (sqrt(Pi) * 3^(n - 1/2) * n^(3/2)). - _Vaclav Kotesovec_, Mar 17 2016
%F A270447 a(n) = [x^n] (1 - sqrt(1 - 4*x))/(2*x*(1 - x)^(n+1)). - _Ilya Gutkovskiy_, Nov 01 2017
%t A270447 Table[Sum[Binomial[2*k,k]/(k+1) * Binomial[2*n-k,n], {k,0,n}], {n,0,25}] (* _Vaclav Kotesovec_, Mar 17 2016 *)
%t A270447 a[n_] := ((2 n + 1) Binomial[2 n, n] (1 - Hypergeometric2F1[-1/2, -n - 1, -2 n - 1, 4]))/(2 (n + 1));
%t A270447 Table[a[n], {n, 0, 24}] (* _Peter Luschny_, May 30 2022 *)
%o A270447 (Maxima)
%o A270447 a(n):=sum((binomial(2*k,k)*binomial(2*n-k,n))/(k+1),k,0,n);
%o A270447 (PARI) a(n) = sum(i=0, n, (binomial(2*i, i)*binomial(2*n-i, n))/(i+1)); \\ _Altug Alkan_, Mar 17 2016
%Y A270447 Cf. A000108, A007317, A092392.
%K A270447 nonn
%O A270447 0,2
%A A270447 _Vladimir Kruchinin_, Mar 17 2016