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A026533 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026519.

%I A026533 #12 Dec 21 2021 02:34:07
%S A026533 1,3,7,18,40,104,231,607,1353,3575,7989,21169,47384,125757,281798,
%T A026533 748638,1678832,4463098,10014074,26635050,59787092,159078450,
%U A026533 357193976,950678416,2135189511,5684158586,12769030254,33999245582
%N A026533 a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A026519.
%H A026533 G. C. Greubel, <a href="/A026533/b026533.txt">Table of n, a(n) for n = 0..1000</a>
%F A026533 a(n) = Sum_{i=0..n} Sum_{j=0..i} A026519(i,j).
%t A026533 T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k], T[n-1, k-1] + T[n-1, k-2] + T[n-1, k] ]]]]; (* T = A026519 *)
%t A026533 a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[i,j], {i,0,n}, {j,0,i}] ];
%t A026533 Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, Dec 20 2021 *)
%o A026533 (Sage)
%o A026533 @CachedFunction
%o A026533 def T(n,k): # T = A026519
%o A026533     if (k<0 or k>2*n): return 0
%o A026533     elif (k==0 or k==2*n): return 1
%o A026533     elif (k==1 or k==2*n-1): return (n+1)//2
%o A026533     elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
%o A026533     else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o A026533 @CachedFunction
%o A026533 def a(n): return sum(sum( T(i,j) for j in (0..i)) for i in (0..n) )
%o A026533 [a(n) for n in (0..40)] # _G. C. Greubel_, Dec 20 2021
%Y A026533 Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026534, A027262, A027263, A027264, A027265, A027266.
%K A026533 nonn
%O A026533 0,2
%A A026533 _Clark Kimberling_