This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A026584 #27 Dec 12 2021 09:30:46
%S A026584 1,1,0,1,1,1,2,1,1,1,1,4,2,4,1,1,1,2,5,7,8,7,5,2,1,1,2,8,9,20,14,20,9,
%T A026584 8,2,1,1,3,9,19,28,43,40,43,28,19,9,3,1,1,3,13,22,56,62,111,86,111,62,
%U A026584 56,22,13,3,1,1,4,14,38,69,140,167,259,222,259,167,140,69,38,14,4,1
%N A026584 Irregular triangular array T read by rows: T(i,0) = T(i,2i) = 1 for i >= 0; T(i,1) = T(i,2i-1) = floor(i/2) for i >= 1; and for i >= 2 and j = 2..2i-2, T(i,j) = T(i-1,j-2) + T(i-1,j-1) + T(i-1,j) if i+j is odd, and T(i,j) = T(i-1,j-2) + T(i-1,j) if i+j is even.
%C A026584 Row sums are in A026597. - _Philippe Deléham_, Oct 16 2006
%C A026584 T(n, k) = number of integer strings s(0)..s(n) such that s(0) = 0, s(n) = n-k, |s(i)-s(i-1)| <= 1 if s(i-1) odd, |s(i)-s(i-1)| = 1 if s(i-1) is even, for i = 1..n.
%H A026584 Clark Kimberling, <a href="/A026584/b026584.txt">Rows 0..100, flattened</a>
%H A026584 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A026584 T(n, k) = T(n-1, k-2) + T(n-1, k) if ( (n+k) mod 2 ) = 0, otherwise T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), where T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor(n/2).
%e A026584 First 5 rows:
%e A026584 1
%e A026584 1 0 1
%e A026584 1 1 2 1 1
%e A026584 1 1 4 2 4 1 1
%e A026584 1 2 5 7 8 7 5 2 1
%t A026584 z = 12; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := Floor[n/2]; t[n_, k_] := Floor[n/2] /; k == 2 n - 1; t[n_, k_] := t[n, k] = If[EvenQ[n + k], t[n - 1, k - 2] + t[n - 1, k], t[n - 1, k - 2] + t[n - 1, k - 1] + t[n - 1, k]]; u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
%t A026584 TableForm[u] (* A026584 array *)
%t A026584 v = Flatten[u] (* A026584 sequence *)
%o A026584 (Sage)
%o A026584 @CachedFunction
%o A026584 def T(n,k):
%o A026584 if (k==0 or k==2*n): return 1
%o A026584 elif (k==1 or k==2*n-1): return (n//2)
%o A026584 else: return T(n-1, k-2) + T(n-1, k) if ((n+k)%2==0) else T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
%o A026584 flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Dec 11 2021
%Y A026584 Cf. A026519, A026536, A026552, A026568, A027926.
%Y A026584 Cf. A026585, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599.
%K A026584 nonn,tabf
%O A026584 1,7
%A A026584 _Clark Kimberling_
%E A026584 Updated by _Clark Kimberling_, Aug 29 2014