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 A193728 #23 Nov 28 2023 16:24:26
%S A193728 1,2,1,8,10,3,32,64,42,9,128,352,360,162,27,512,1792,2496,1728,594,81,
%T A193728 2048,8704,15360,14400,7560,2106,243,8192,40960,87552,103680,73440,
%U A193728 31104,7290,729,32768,188416,473088,677376,604800,344736,122472,24786,2187
%N A193728 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (x+2)^n and q(n,x) = (2*x+1)^n.
%C A193728 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%C A193728 Triangle T(n,k), read by rows, given by (2,2,0,0,0,0,0,0,0,...) DELTA (1,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 05 2011
%H A193728 G. C. Greubel, <a href="/A193728/b193728.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A193728 T(n,k) = 3*T(n-1,k-1) + 4*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - _Philippe Deléham_, Oct 05 2011
%F A193728 G.f.: (1-2*x-2*x*y)/(1-4*x-3*x*y). - _R. J. Mathar_, Aug 11 2015
%F A193728 From _G. C. Greubel_, Nov 28 2023: (Start)
%F A193728 T(n, n-k) = A193729(n, k).
%F A193728 T(n, 0) = A081294(n).
%F A193728 T(n, n-1) = 2*A081038(n-1).
%F A193728 T(n, n) = A133494(n).
%F A193728 Sum_{k=0..n} T(n, k) = (1/7)*(4*[n=0] + 3*A000420(n)).
%F A193728 Sum_{k=0..n} (-1)^k * T(n, k) = A000012(n).
%F A193728 Sum_{k=0..floor(n/2)} T(n-k, k) = (5*b(n) + 4*b(n-1))/14 + (2/3)*[n=0].
%F A193728 Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A060816(n),
%F A193728 where b(n) = (2 + sqrt(7))^n + (2 - sqrt(7))^n. (End)
%e A193728 First six rows:
%e A193728 1;
%e A193728 2, 1;
%e A193728 8, 10, 3;
%e A193728 32, 64, 42, 9;
%e A193728 128, 352, 360, 162, 27;
%e A193728 512, 1792, 2496, 1728, 594, 81;
%t A193728 (* First program *)
%t A193728 z = 8; a = 1; b = 2; c = 2; d = 1;
%t A193728 p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
%t A193728 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193728 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193728 g[n_] := CoefficientList[w[n, x], {x}]
%t A193728 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193728 Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193728 *)
%t A193728 TableForm[Table[g[n], {n, -1, z}]]
%t A193728 Flatten[Table[g[n], {n, -1, z}]] (* A193729 *)
%t A193728 (* Second program *)
%t A193728 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 4*T[n-1,k] + 3*T[n-1,k-1]]];
%t A193728 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 28 2023 *)
%o A193728 (Magma)
%o A193728 function T(n, k) // T = A193728
%o A193728 if k lt 0 or k gt n then return 0;
%o A193728 elif n lt 2 then return n-k+1;
%o A193728 else return 4*T(n-1, k) + 3*T(n-1, k-1);
%o A193728 end if;
%o A193728 end function;
%o A193728 [T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 28 2023
%o A193728 (SageMath)
%o A193728 def T(n, k): # T = A193728
%o A193728 if (k<0 or k>n): return 0
%o A193728 elif (n<2): return n-k+1
%o A193728 else: return 4*T(n-1, k) + 3*T(n-1, k-1)
%o A193728 flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Nov 28 2023
%Y A193728 Cf. A000012, A000420, A060816, A081038, A081294.
%Y A193728 Cf. A084938, A133494, A193722, A193729.
%K A193728 nonn,tabl
%O A193728 0,2
%A A193728 _Clark Kimberling_, Aug 04 2011