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A026566 a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=n, T given by A026552.

%I A026566 #10 Dec 19 2021 04:30:41
%S A026566 1,3,9,20,53,117,308,684,1806,4028,10664,23844,63239,141612,376026,
%T A026566 842866,2239900,5024166,13359408,29980384,79753402,179044760,
%U A026566 476451644,1069936084,2847931619,6396900694,17030741437,38260956765
%N A026566 a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=n, T given by A026552.
%H A026566 G. C. Greubel, <a href="/A026566/b026566.txt">Table of n, a(n) for n = 0..1000</a>
%F A026566 a(n) = Sum_{i=0..n} Sum_{j=0..i} A026552(i, j).
%t A026566 T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/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=A026552 *)
%t A026566 a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[i,j], {i,0,n}, {j,0,i}]];
%t A026566 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Dec 19 2021 *)
%o A026566 (Sage)
%o A026566 @CachedFunction
%o A026566 def T(n,k): # T = A026552
%o A026566     if (k==0 or k==2*n): return 1
%o A026566     elif (k==1 or k==2*n-1): return (n+2)//2
%o A026566     elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o A026566     else: return T(n-1, k) + T(n-1, k-2)
%o A026566 @CachedFunction
%o A026566 def a(n): return sum( sum( T(i,j) for j in (0..i) ) for i in (0..n) )
%o A026566 [a(n) for n in (0..40)] # _G. C. Greubel_, Dec 19 2021
%Y A026566 Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026567, A027272, A027273, A027274, A027275, A027276.
%K A026566 nonn
%O A026566 0,2
%A A026566 _Clark Kimberling_