%I #12 Oct 30 2022 09:00:40
%S 1,1,2,3,3,4,7,9,14,23,33,52,85,127,202,329,503,804,1307,2027,3250,
%T 5277,8263,13276,21539,33957,54638,88595,140373,226108,366481,582865,
%U 939622,1522487,2428517,3917412,6345929,10145769,16374126
%N a(n) = Sum_{i=0..floor(n/2)} A047072(i, n-2*i).
%H G. C. Greubel, <a href="/A047079/b047079.txt">Table of n, a(n) for n = 0..1000</a>
%t T[n_, k_]:= T[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 A047079[n_]:= Sum[T[j, n-2*j], {j,0,Floor[n/2]}] +Boole[n==0];
%t Table[A047079[n], {n,0,50}] (* _G. C. Greubel_, Oct 29 2022 *)
%o (Magma)
%o b:= func< n | n eq 0 select 1 else 2*Catalan(n-1) >;
%o function A(n, k)
%o if k eq n then return b(n);
%o elif k gt n then return Binomial(n+k-1, n) - Binomial(n+k-1, n-1);
%o else return Binomial(n+k-1, k) - Binomial(n+k-1, k-1);
%o end if; return A;
%o end function;
%o [(&+[A(j, n-2*j): j in [0..Floor(n/2)]]): n in [0..50]]; // _G. C. Greubel_, Oct 29 2022
%o (SageMath)
%o def A047072(n, k): # array
%o if (k==n): return 2*catalan_number(n-1) + 2*int(n==0)
%o elif (k>n): return binomial(n+k-1, n) - binomial(n+k-1, n-1)
%o else: return binomial(n+k-1, k) - binomial(n+k-1, k-1)
%o def A047079(n): return sum( A047072(j, n-2*j) for j in range(((n+1)//2)+1) )
%o [A047079(n) for n in range(51)] # _G. C. Greubel_, Oct 29 2022
%Y Cf. A047072, A047073, A047074.
%K nonn
%O 0,3
%A _Clark Kimberling_
%E Name improved by _Sean A. Irvine_, May 11 2021