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A026520 a(n) = T(n,n), T given by A026519. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 0.

%I A026520 #22 Dec 20 2021 09:43:18
%S A026520 1,1,2,4,8,20,38,104,196,556,1052,3032,5774,16778,32146,93872,180772,
%T A026520 529684,1024256,3008864,5837908,17184188,33433996,98577712,192239854,
%U A026520 567591142,1109049320,3278348608,6416509142,18986482250
%N A026520 a(n) = T(n,n), T given by A026519. Also a(n) = number of integer strings s(0), ..., s(n), counted by T, such that s(n) = 0.
%H A026520 G. C. Greubel, <a href="/A026520/b026520.txt">Table of n, a(n) for n = 0..1000</a>
%H A026520 Veronika Irvine, <a href="http://hdl.handle.net/1828/7495">Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns</a>, PhD Dissertation, University of Victoria, 2016.
%H A026520 Veronika Irvine, Stephen Melczer, and Frank Ruskey, <a href="https://arxiv.org/abs/1804.08725">Vertically constrained Motzkin-like paths inspired by bobbin lace</a>, arXiv:1804.08725 [math.CO], 2018.
%F A026520 a(n) = A026519(n, n).
%F A026520 For n>1, a(n) = 2*A026554(n-1).
%t A026520 T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/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 = A026519 *)
%t A026520 Table[T[n, n], {n,0,40}] (* _G. C. Greubel_, Dec 19 2021 *)
%o A026520 (Sage)
%o A026520 @CachedFunction
%o A026520 def T(n,k): # T = A026552
%o A026520     if (k==0 or k==2*n): return 1
%o A026520     elif (k==1 or k==2*n-1): return (n+1)//2
%o A026520     elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
%o A026520     else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o A026520 [T(n,n) for n in (0..40)] # _G. C. Greubel_, Dec 19 2021
%Y A026520 Cf. A026519, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.
%K A026520 nonn
%O A026520 0,3
%A A026520 _Clark Kimberling_