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 A193735 #25 Nov 22 2023 01:45:12
%S A193735 1,2,1,8,6,1,32,32,10,1,128,160,72,14,1,512,768,448,128,18,1,2048,
%T A193735 3584,2560,960,200,22,1,8192,16384,13824,6400,1760,288,26,1,32768,
%U A193735 73728,71680,39424,13440,2912,392,30,1,131072,327680,360448,229376,93184,25088,4480,512,34,1
%N A193735 Mirror of the triangle A193734.
%C A193735 A193735 is obtained by reversing the rows of the triangle A193734.
%C A193735 Triangle T(n,k), read by rows, given by (2,2,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 05 2011
%H A193735 G. C. Greubel, <a href="/A193735/b193735.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A193735 T(n,k) = A193734(n,n-k).
%F A193735 T(n,k) = 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 A193735 G.f.: (1-2*x)/(1-4*x-x*y). - _R. J. Mathar_, Aug 11 2015
%F A193735 From _G. C. Greubel_, Nov 19 2023: (Start)
%F A193735 T(n, 0) = A081294(n).
%F A193735 Sum_{k=0..n} T(n, k) = A005053(n).
%F A193735 Sum_{k=0..n} (-1)^k * T(n, k) = A133494(n).
%F A193735 Sum_{k=0..floor(n/2)} T(n-k, k) = A001077(n).
%F A193735 Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A001075(n). (End)
%e A193735 First six rows:
%e A193735 1;
%e A193735 2, 1;
%e A193735 8, 6, 1;
%e A193735 32, 32, 10, 1;
%e A193735 128, 160, 72, 14, 1;
%e A193735 512, 768, 448, 128, 18, 1;
%t A193735 (* First program *)
%t A193735 z = 8; a = 2; b = 1; c = 1; d = 2;
%t A193735 p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
%t A193735 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193735 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193735 g[n_] := CoefficientList[w[n, x], {x}]
%t A193735 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193735 Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193734 *)
%t A193735 TableForm[Table[g[n], {n, -1, z}]]
%t A193735 Flatten[Table[g[n], {n, -1, z}]] (* A193735 *)
%t A193735 (* Second program *)
%t A193735 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 4*T[n-1, k] + T[n -1, k-1]]];
%t A193735 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 19 2023 *)
%o A193735 (Magma)
%o A193735 function T(n, k) // T = A193735
%o A193735 if k lt 0 or k gt n then return 0;
%o A193735 elif n lt 2 then return n-k+1;
%o A193735 else return 4*T(n-1, k) + T(n-1, k-1);
%o A193735 end if;
%o A193735 end function;
%o A193735 [T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 19 2023
%o A193735 (SageMath)
%o A193735 def T(n, k): # T = A193735
%o A193735 if (k<0 or k>n): return 0
%o A193735 elif (n<2): return n-k+1
%o A193735 else: return 4*T(n-1, k) + T(n-1, k-1)
%o A193735 flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Nov 19 2023
%Y A193735 Cf. A084938, A193722, A193734.
%Y A193735 Cf. A001075, A001077, A005053, A081294, A133494.
%K A193735 nonn,tabl
%O A193735 0,2
%A A193735 _Clark Kimberling_, Aug 04 2011