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A047079 a(n) = Sum_{i=0..floor(n/2)} A047072(i, n-2*i).

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