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 A026626 #17 Jun 20 2024 11:08:24
%S A026626 1,1,1,1,3,1,1,4,4,1,1,6,8,6,1,1,7,14,14,7,1,1,9,21,28,21,9,1,1,10,30,
%T A026626 49,49,30,10,1,1,12,40,79,98,79,40,12,1,1,13,52,119,177,177,119,52,13,
%U A026626 1,1,15,65,171,296,354,296,171,65,15,1,1,16,80,236,467,650,650,467,236,80,16,1
%N A026626 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/2) for n >= 1, T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2, n >= 4.
%H A026626 G. C. Greubel, <a href="/A026626/b026626.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A026626 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A026626 T(n, k) = number of paths from (0, 0) to (n-k, k) in the 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 >= 0 and even and for j=0, i >= 0 and even.
%F A026626 From _G. C. Greubel_, Jun 19 2024: (Start)
%F A026626 T(n, n-k) = T(n, k)
%F A026626 T(n, 1) = (-1)^n*A084056(n) = A032766(n), n >= 1.
%F A026626 T(n, 2) = A006578(n-1), n >= 2.
%F A026626 T(n, 3) = (1/16)*(4*n^3 - 14*n^2 + 12*n + 15 + (-1)^n) - [n=3] , n >= 3.
%F A026626 Sum_{k=0..n} (-1)^k*T(n, k) = A176742(n) + [n=2].
%F A026626 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1/4)*((-1)^n*(8/sqrt(3)* sin(2*(n+1)*Pi/3) - 2*cos(n*Pi/2) + 1) - 3) + [n<2].
%F A026626 Sum_{k=0..n} k*T(n, k) = (1/6)*n*(17*2^(n-2) - 2 - (1-(-1)^n)) + (1/4)*[n=1]. (End)
%e A026626 Triangle begins as:
%e A026626 1;
%e A026626 1, 1;
%e A026626 1, 3, 1;
%e A026626 1, 4, 4, 1;
%e A026626 1, 6, 8, 6, 1;
%e A026626 1, 7, 14, 14, 7, 1;
%e A026626 1, 9, 21, 28, 21, 9, 1;
%e A026626 1, 10, 30, 49, 49, 30, 10, 1;
%e A026626 1, 12, 40, 79, 98, 79, 40, 12, 1;
%e A026626 1, 13, 52, 119, 177, 177, 119, 52, 13, 1;
%e A026626 1, 15, 65, 171, 296, 354, 296, 171, 65, 15, 1;
%t A026626 T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, (6*n-1+(-1)^n)/4, T[n-1, k-1] + T[n-1, k] ]];
%t A026626 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 19 2024 *)
%o A026626 (Magma)
%o A026626 function T(n,k) // T = A026626
%o A026626 if k eq 0 or k eq n then return 1;
%o A026626 elif k eq 1 or k eq n-1 then return Floor(3*n/2);
%o A026626 else return T(n-1,k-1) + T(n-1,k);
%o A026626 end if;
%o A026626 end function;
%o A026626 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 19 2024
%o A026626 (SageMath)
%o A026626 def T(n,k): # T = A026626
%o A026626 if (k==0 or k==n): return 1
%o A026626 elif (k==1 or k==n-1): return int(3*n//2)
%o A026626 else: return T(n-1,k-1) + T(n-1,k)
%o A026626 flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jun 19 2024
%Y A026626 Cf. A006578, A026627, A026628, A026629, A026630, A026631, A026632.
%Y A026626 Cf. A026633, A026634, A026635, A026636, A026961, A026962, A026963.
%Y A026626 Cf. A026964, A026965, A032766, A084056.
%K A026626 nonn,tabl,easy
%O A026626 0,5
%A A026626 _Clark Kimberling_