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A026563 a(n) = T(n, floor(n/2)), where T is given by A026552.

%I A026563 #8 Dec 18 2021 04:01:30
%S A026563 1,1,2,2,7,8,24,28,93,111,362,436,1452,1763,5880,7176,24089,29521,
%T A026563 99386,122182,412637,508595,1721500,2126312,7211536,8923136,30312960,
%U A026563 37563930,127790379,158563368,540082784,670893296,2287577537
%N A026563 a(n) = T(n, floor(n/2)), where T is given by A026552.
%H A026563 G. C. Greubel, <a href="/A026563/b026563.txt">Table of n, a(n) for n = 0..1000</a>
%F A026563 a(n) = A026552(n, floor(n/2)).
%t A026563 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 A026563 a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, T[n, Floor[n/2]]];
%t A026563 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Dec 18 2021 *)
%o A026563 (Sage)
%o A026563 @CachedFunction
%o A026563 def T(n,k): # T = A026552
%o A026563     if (k==0 or k==2*n): return 1
%o A026563     elif (k==1 or k==2*n-1): return (n+2)//2
%o A026563     elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o A026563     else: return T(n-1, k) + T(n-1, k-2)
%o A026563 [T(n,n//2) for n in (0..40)] # _G. C. Greubel_, Dec 18 2021
%Y A026563 Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026566, A026567, A027272, A027273, A027274, A027275, A027276.
%K A026563 nonn
%O A026563 0,3
%A A026563 _Clark Kimberling_