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 A193730 #24 Nov 21 2023 17:12:06
%S A193730 1,2,1,4,8,3,8,28,30,9,16,80,144,108,27,32,208,528,648,378,81,64,512,
%T A193730 1680,2880,2700,1296,243,128,1216,4896,10800,14040,10692,4374,729,256,
%U A193730 2816,13440,36288,60480,63504,40824,14580,2187,512,6400,35328,112896,229824,308448,272160,151632,48114,6561
%N A193730 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (2x+1)^n and q(n,x) = (2x+1)^n.
%C A193730 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%C A193730 Triangle T(n,k), read by rows, given by (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 A193730 G. C. Greubel, <a href="/A193730/b193730.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A193730 T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - _Philippe Deléham_, Oct 05 2011
%F A193730 G.f.: (1-2*x*y)/(1-2*x-3*x*y). - _R. J. Mathar_, Aug 11 2015
%F A193730 From _G. C. Greubel_, Nov 20 2023: (Start)
%F A193730 T(n, 0) = A000079(n).
%F A193730 T(n, 1) = A130129(n-1).
%F A193730 T(n, n) = A133494(n).
%F A193730 T(n, n-1) = A199923(n).
%F A193730 Sum_{k=0..n} T(n, k) = A005053(n).
%F A193730 Sum_{k=0..n} (-1)^k * T(n, k) = A165326(n). (End)
%e A193730 First six rows:
%e A193730 1;
%e A193730 2, 1;
%e A193730 4, 8, 3;
%e A193730 8, 28, 30, 9;
%e A193730 16, 80, 144, 108, 27;
%e A193730 32, 208, 528, 648, 378, 81;
%t A193730 (* First program *)
%t A193730 z = 8; a = 2; b = 1; c = 2; d = 1;
%t A193730 p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
%t A193730 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193730 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193730 g[n_] := CoefficientList[w[n, x], {x}]
%t A193730 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193730 Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193730 *)
%t A193730 TableForm[Table[g[n], {n, -1, z}]]
%t A193730 Flatten[Table[g[n], {n, -1, z}]] (* A193731 *)
%t A193730 (* Second program *)
%t A193730 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 2*T[n-1, k] + 3*T[n-1, k-1]]];
%t A193730 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 20 2023 *)
%o A193730 (Magma)
%o A193730 function T(n, k) // T = A193730
%o A193730 if k lt 0 or k gt n then return 0;
%o A193730 elif n lt 2 then return n-k+1;
%o A193730 else return 2*T(n-1, k) + 3*T(n-1, k-1);
%o A193730 end if;
%o A193730 end function;
%o A193730 [T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2023
%o A193730 (SageMath)
%o A193730 def T(n, k): # T = A193730
%o A193730 if (k<0 or k>n): return 0
%o A193730 elif (n<2): return n-k+1
%o A193730 else: return 2*T(n-1, k) + 3*T(n-1, k-1)
%o A193730 flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Nov 20 2023
%Y A193730 Cf. A000079, A005053, A084938, A130129, A133494, A165326.
%Y A193730 Cf. A193722, A193731, A199923.
%K A193730 nonn,tabl
%O A193730 0,2
%A A193730 _Clark Kimberling_, Aug 04 2011