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 A193729 #27 Nov 28 2023 15:56:55
%S A193729 1,1,2,3,10,8,9,42,64,32,27,162,360,352,128,81,594,1728,2496,1792,512,
%T A193729 243,2106,7560,14400,15360,8704,2048,729,7290,31104,73440,103680,
%U A193729 87552,40960,8192,2187,24786,122472,344736,604800,677376,473088,188416,32768
%N A193729 Mirror of the triangle A193728.
%C A193729 T(n, k) is obtained by reversing the rows of the triangle A193728.
%C A193729 Triangle T(n,k), read by rows, given by [1,2,0,0,0,0,...] DELTA [2,2,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 05 2011
%H A193729 G. C. Greubel, <a href="/A193729/b193729.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A193729 Let w(n,k) be the triangle of A193728, then the triangle in this sequence is given by w(n,n-k).
%F A193729 T(n,k) = 4*T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - _Philippe Deléham_, Oct 05 2011
%F A193729 G.f.: (1-2*x-2*x*y)/(1-3*x-4*x*y). - _R. J. Mathar_, Aug 11 2015
%F A193729 From _G. C. Greubel_, Nov 28 2023: (Start)
%F A193729 T(n, 0) = A133494(n).
%F A193729 T(n, 1) = 2*A081038(n-1).
%F A193729 T(n, n) = A081294(n).
%F A193729 Sum_{k=0..n} T(n, k) = (1/7)*(4*[n=0] + 3*A000420(n)).
%F A193729 Sum_{k=0..n} (-1)^k * T(n, k) = A033999(n).
%F A193729 Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*[n=0] + A108981(n-1).
%F A193729 Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = (1/2)*[n=0] + A247560(n-1).
%F A193729 (End)
%e A193729 First six rows:
%e A193729 1;
%e A193729 1, 2;
%e A193729 3, 10, 8;
%e A193729 9, 42, 64, 32;
%e A193729 27, 162, 360, 352, 128;
%e A193729 81, 594, 1728, 2496, 1792, 512;
%t A193729 (* First program *)
%t A193729 z = 8; a = 1; b = 2; c = 2; d = 1;
%t A193729 p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
%t A193729 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193729 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193729 g[n_] := CoefficientList[w[n, x], {x}]
%t A193729 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193729 Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193728 *)
%t A193729 TableForm[Table[g[n], {n, -1, z}]]
%t A193729 Flatten[Table[g[n], {n, -1, z}]] (* A193729 *)
%t A193729 (* Second program *)
%t A193729 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 3*T[n-1,k] + 4*T[n -1, k-1]]];
%t A193729 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 28 2023 *)
%o A193729 (Magma)
%o A193729 function T(n, k) // T = A193729
%o A193729 if k lt 0 or k gt n then return 0;
%o A193729 elif n lt 2 then return k+1;
%o A193729 else return 3*T(n-1, k) + 4*T(n-1, k-1);
%o A193729 end if;
%o A193729 end function;
%o A193729 [T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 28 2023
%o A193729 (SageMath)
%o A193729 def T(n, k): # T = A193729
%o A193729 if (k<0 or k>n): return 0
%o A193729 elif (n<2): return k+1
%o A193729 else: return 3*T(n-1, k) + 4*T(n-1, k-1)
%o A193729 flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Nov 28 2023
%Y A193729 Cf. A000420, A033999, A081038, A081294, A084938.
%Y A193729 Cf. A108981, A133494, A193722, A193728, A247560.
%K A193729 nonn,tabl
%O A193729 0,3
%A A193729 _Clark Kimberling_, Aug 04 2011