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%I #10 Dec 17 2021 22:15:01
%S 1,1,3,4,12,20,58,104,300,556,1608,3032,8806,16778,48924,93872,274644,
%T 529684,1553940,3008864,8846772,17184188,50618184,98577712,290817566,
%U 567591142,1676640462,3278348608,9694857750,18986482250
%N a(n) = T(n,n), T given by A026552. Also a(n) is the number of integer strings s(0),...,s(n) counted by T, such that s(n)=0.
%H G. C. Greubel, <a href="/A026553/b026553.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = A026552(n,n).
%t 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 Table[T[n,n], {n,0,40}] (* _G. C. Greubel_, Dec 17 2021 *)
%o (Sage)
%o @CachedFunction
%o def T(n,k): # T = A026552
%o if (k==0 or k==2*n): return 1
%o elif (k==1 or k==2*n-1): return (n+2)//2
%o elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o else: return T(n-1, k) + T(n-1, k-2)
%o [T(n,n) for n in (0..40)] # _G. C. Greubel_, Dec 17 2021
%Y Cf. A026552, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.
%K nonn
%O 0,3
%A _Clark Kimberling_
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