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A026537 a(n) = T(n,n), T given by A026536. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n)=0.

%I A026537 #9 Apr 11 2022 14:57:32
%S A026537 1,0,2,2,8,12,38,66,196,360,1052,1980,5774,11004,32146,61726,180772,
%T A026537 348912,1024256,1984608,5837908,11346280,33433996,65143716,192239854,
%U A026537 375351288,1109049320,2169299288,6416509142,12569973108
%N A026537 a(n) = T(n,n), T given by A026536. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n)=0.
%H A026537 G. C. Greubel, <a href="/A026537/b026537.txt">Table of n, a(n) for n = 0..1000</a>
%F A026537 a(n) = A026536(n, n).
%F A026537 a(n) = 2 * A026521(n-1).
%t A026537 T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[n/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k-1] + T[n-1, k], T[n-1, k-2] + T[n-1, k]] ]]; Table[T[n, n], {n, 0, 35}] (* _G. C. Greubel_, Apr 10 2022 *)
%o A026537 (SageMath)
%o A026537 @cached_function
%o A026537 def T(n, k): # A026536
%o A026537     if k < 0 or n < 0: return 0
%o A026537     elif k == 0 or k == 2*n: return 1
%o A026537     elif k == 1 or k == 2*n-1: return n//2
%o A026537     elif n % 2 == 1: return T(n-1, k-2) + T(n-1, k)
%o A026537     return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
%o A026537 def A026537(n): return T(n,n)
%o A026537 [A026537(n) for n in (0..35)] # _G. C. Greubel_, Apr 10 2022
%Y A026537 Cf. A026521, A026536.
%K A026537 nonn
%O A026537 0,3
%A A026537 _Clark Kimberling_