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A026554 a(n) = T(n,n-1), T given by A026552. Also a(n) is the number of integer strings s(0),...,s(n) counted by T, such that s(n)=1.

%I A026554 #12 Dec 17 2021 20:55:43
%S A026554 1,2,4,10,19,52,98,278,526,1516,2887,8389,16073,46936,90386,264842,
%T A026554 512128,1504432,2918954,8592094,16716998,49288856,96119927,283795571,
%U A026554 554524660,1639174304,3208254571,9493241125,18607536319,55108565584
%N A026554 a(n) = T(n,n-1), T given by A026552. Also a(n) is the number of integer strings s(0),...,s(n) counted by T, such that s(n)=1.
%H A026554 G. C. Greubel, <a href="/A026554/b026554.txt">Table of n, a(n) for n = 1..1000</a>
%F A026554 a(n) = A026520(n+1)/2.
%t A026554 T[n_, k_]:= T[n, k]= If[k>2*n, 0, 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 A026554 Table[T[n, n-1], {n, 40}] (* _G. C. Greubel_, Dec 17 2021 *)
%o A026554 (Sage)
%o A026554 @CachedFunction
%o A026554 def T(n,k): # T = A026552
%o A026554     if (k==0 or k==2*n): return 1
%o A026554     elif (k==1 or k==2*n-1): return (n+2)//2
%o A026554     elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o A026554     else: return T(n-1, k) + T(n-1, k-2)
%o A026554 [T(n,n-1) for n in (1..40)] # _G. C. Greubel_, Dec 17 2021
%Y A026554 Cf. A026520.
%Y A026554 Cf. A026552, A026553, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026563, A026566, A026567, A027272, A027273, A027274, A027275, A027276.
%K A026554 nonn
%O A026554 1,2
%A A026554 _Clark Kimberling_