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A047074 a(n) = Sum_{i=0..floor(n/2)} T(i,n-i), array T as in A047072.

%I A047074 #15 Oct 30 2022 09:00:50
%S A047074 1,1,3,2,5,6,14,20,45,70,154,252,546,924,1980,3432,7293,12870,27170,
%T A047074 48620,102102,184756,386308,705432,1469650,2704156,5616324,10400600,
%U A047074 21544100,40116600,82907640,155117520,319929885,601080390,1237518450
%N A047074 a(n) = Sum_{i=0..floor(n/2)} T(i,n-i), array T as in A047072.
%H A047074 G. C. Greubel, <a href="/A047074/b047074.txt">Table of n, a(n) for n = 0..1000</a>
%t A047074 A[n_, k_]:= A[n, k]= If[k==n, 2*CatalanNumber[n-1] +2*Boole[n==0], If[k>n, Binomial[n+k-1,n] -Binomial[n+k-1,n-1], Binomial[n+k-1,k] - Binomial[n+k-1, k - 1]]];
%t A047074 A047074[n_]:= Sum[A[j, n-j], {j,0,Floor[n/2]}] +Boole[n==0];
%t A047074 Table[A047074[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 29 2022 *)
%o A047074 (Magma)
%o A047074 b:= func< n | n eq 0 select 1 else 2*Catalan(n-1) >;
%o A047074 function A(n, k)
%o A047074   if k eq n then return b(n);
%o A047074   elif k gt n then return Binomial(n+k-1, n) - Binomial(n+k-1, n-1);
%o A047074   else return Binomial(n+k-1, k) - Binomial(n+k-1, k-1);
%o A047074   end if; return A;
%o A047074 end function;
%o A047074 [(&+[A(j, n-j): j in [0..Floor(n/2)]]): n in [0..50]]; // _G. C. Greubel_, Oct 29 2022
%o A047074 (SageMath)
%o A047074 def A047072(n, k): # array
%o A047074     if (k==n): return 2*catalan_number(n-1) + 2*int(n==0)
%o A047074     elif (k>n): return binomial(n+k-1, n) - binomial(n+k-1, n-1)
%o A047074     else: return binomial(n+k-1, k) - binomial(n+k-1, k-1)
%o A047074 def A047074(n): return sum( A047072(j, n-j) for j in range((n//2)+1) )
%o A047074 [A047074(n) for n in range(51)] # _G. C. Greubel_, Oct 29 2022
%Y A047074 Cf. A047072, A047073, A047079.
%K A047074 nonn
%O A047074 0,3
%A A047074 _Clark Kimberling_
%E A047074 Extra leading 1 removed by _Sean A. Irvine_, May 11 2021