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 A026637 #31 Jun 30 2024 09:10:04
%S A026637 1,1,1,1,2,1,1,4,4,1,1,5,8,5,1,1,7,13,13,7,1,1,8,20,26,20,8,1,1,10,28,
%T A026637 46,46,28,10,1,1,11,38,74,92,74,38,11,1,1,13,49,112,166,166,112,49,13,
%U A026637 1,1,14,62,161,278,332,278,161,62,14,1,1,16,76,223,439,610,610,439,223,76,16,1
%N A026637 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,1) = T(n,n-1) = floor((3*n-1)/2) for n >= 1, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2, n >= 4.
%C A026637 T(n, k) = number of paths from (0, 0) to (n-k, k) in directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, j)-to-(i+1, j+1) for i=0, j >= 1 and odd and for j=0, i >= 1 and odd.
%C A026637 See A228053 for a sequence with many terms in common with this one. - _T. D. Noe_, Aug 07 2013
%H A026637 Reinhard Zumkeller, <a href="/A026637/b026637.txt">Rows n = 0..100 of table, flattened</a>
%F A026637 From _G. C. Greubel_, Jun 28 2024: (Start)
%F A026637 T(n, n-k) = T(n, k).
%F A026637 T(2*n-1, n-1) = A026641(n), n >= 1.
%F A026637 Sum_{k=0..n} T(n, k) = A026644(n).
%F A026637 Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)
%e A026637 Triangle begins as:
%e A026637 1;
%e A026637 1, 1;
%e A026637 1, 2, 1;
%e A026637 1, 4, 4, 1;
%e A026637 1, 5, 8, 5, 1;
%e A026637 1, 7, 13, 13, 7, 1;
%e A026637 1, 8, 20, 26, 20, 8, 1;
%e A026637 1, 10, 28, 46, 46, 28, 10, 1;
%e A026637 1, 11, 38, 74, 92, 74, 38, 11, 1;
%e A026637 1, 13, 49, 112, 166, 166, 112, 49, 13, 1;
%e A026637 1, 14, 62, 161, 278, 332, 278, 161, 62, 14, 1;
%p A026637 A026637 := proc(n,k)
%p A026637 option remember;
%p A026637 if k=0 or k=n then
%p A026637 1
%p A026637 elif k=1 or k=n-1 then
%p A026637 floor((3*n-1)/2) ;
%p A026637 elif k <0 or k > n then
%p A026637 0;
%p A026637 else
%p A026637 procname(n-1,k-1)+procname(n-1,k) ;
%p A026637 end if;
%p A026637 end proc: # _R. J. Mathar_, Apr 26 2015
%t A026637 T[n_, k_] := T[n, k] = Which[k == 0 || k == n, 1, k == 1 || k == n-1, Floor[(3n-1)/2], k < 0 || k > n, 0, True, T[n-1, k-1] + T[n-1, k]];
%t A026637 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 30 2018 *)
%o A026637 (Haskell)
%o A026637 a026637 n k = a026637_tabl !! n !! k
%o A026637 a026637_row n = a026637_tabl !! n
%o A026637 a026637_tabl = [1] : [1,1] : map (fst . snd)
%o A026637 (iterate f (0, ([1,2,1], [0,1,1,0]))) where
%o A026637 f (i, (xs, ws)) = (1 - i,
%o A026637 if i == 1 then (ys, ws) else (zipWith (+) ys ws, ws'))
%o A026637 where ys = zipWith (+) ([0] ++ xs) (xs ++ [0])
%o A026637 ws' = [0,1,0,0] ++ drop 2 ws
%o A026637 -- _Reinhard Zumkeller_, Aug 08 2013
%o A026637 (Magma)
%o A026637 function T(n,k) // T = A026637
%o A026637 if k eq 0 or k eq n then return 1;
%o A026637 elif k eq 1 or k eq n-1 then return Floor((3*n-1)/2);
%o A026637 else return T(n-1, k) + T(n-1, k-1);
%o A026637 end if;
%o A026637 end function;
%o A026637 [T(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jun 28 2024
%o A026637 (SageMath)
%o A026637 def T(n,k): # T = A026637
%o A026637 if k==0 or k==n: return 1
%o A026637 elif k==1 or k==n-1: return ((3*n-1)//2)
%o A026637 else: return T(n-1, k) + T(n-1, k-1)
%o A026637 flatten([[T(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Jun 28 2024
%Y A026637 Cf. A026638, A026639, A026640, A026641, A026642, A026643, A026644.
%Y A026637 Cf. A026966, A026967, A026968, A026969, A026970, A228053.
%Y A026637 Sums include: A000007 (alternating sign row), A026644 (row), A026645, A026646, A026647 (diagonal).
%K A026637 nonn,tabl,easy
%O A026637 0,5
%A A026637 _Clark Kimberling_