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 A193731 #26 Nov 22 2023 01:45:01
%S A193731 1,1,2,3,8,4,9,30,28,8,27,108,144,80,16,81,378,648,528,208,32,243,
%T A193731 1296,2700,2880,1680,512,64,729,4374,10692,14040,10800,4896,1216,128,
%U A193731 2187,14580,40824,63504,60480,36288,13440,2816,256,6561,48114,151632,272160,308448,229824,112896,35328,6400,512
%N A193731 Mirror of the triangle A193730.
%C A193731 A193731 is obtained by reversing the rows of the triangle A193730.
%C A193731 Triangle T(n,k), read by rows, given by (1,2,0,0,0,0,0,0,0,...) DELTA (2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 05 2011
%H A193731 G. C. Greubel, <a href="/A193731/b193731.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A193731 T(n,k) = A193730(n,n-k).
%F A193731 T(n,k) = 2*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 A193731 G.f.: (1-2*x)/(1-3*x-2*x*y). - _R. J. Mathar_, Aug 11 2015
%F A193731 From _G. C. Greubel_, Nov 20 2023: (Start)
%F A193731 T(n, 0) = A133494(n).
%F A193731 T(n, 1) = 2*A006234(n+2).
%F A193731 T(n, 2) = 4*A080420(n-2).
%F A193731 T(n, 3) = 8*A080421(n-3).
%F A193731 T(n, 4) = 16*A080422(n-4).
%F A193731 T(n, 5) = 32*A080423(n-5).
%F A193731 T(n, n) = A000079(n).
%F A193731 T(n, n-1) = A130129(n-1).
%F A193731 Sum_{k=0..n} T(n, k) = A005053(n).
%F A193731 Sum_{k=0..n} (-1)^k * T(n, k) = A153881(n).
%F A193731 Sum_{k=0..floor(n/2)} T(n-k, k) = A007483(n-1).
%F A193731 Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A000012(n). (End)
%e A193731 First six rows:
%e A193731 1;
%e A193731 1, 2;
%e A193731 3, 8, 4;
%e A193731 9, 30, 28, 8;
%e A193731 27, 108, 144, 80, 16;
%e A193731 81, 378, 648, 528, 208, 32;
%t A193731 (* First program *)
%t A193731 z = 8; a = 2; b = 1; c = 2; d = 1;
%t A193731 p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
%t A193731 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193731 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193731 g[n_] := CoefficientList[w[n, x], {x}]
%t A193731 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193731 Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193730 *)
%t A193731 TableForm[Table[g[n], {n, -1, z}]]
%t A193731 Flatten[Table[g[n], {n, -1, z}]] (* A193731 *)
%t A193731 (* Second program *)
%t A193731 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 3*T[n-1, k] + 2*T[n -1, k-1]]];
%t A193731 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 20 2023 *)
%o A193731 (Magma)
%o A193731 function T(n, k) // T = A193731
%o A193731 if k lt 0 or k gt n then return 0;
%o A193731 elif n lt 2 then return k+1;
%o A193731 else return 3*T(n-1, k) + 2*T(n-1, k-1);
%o A193731 end if;
%o A193731 end function;
%o A193731 [T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2023
%o A193731 (SageMath)
%o A193731 def T(n, k): # T = A193731
%o A193731 if (k<0 or k>n): return 0
%o A193731 elif (n<2): return k+1
%o A193731 else: return 3*T(n-1, k) + 2*T(n-1, k-1)
%o A193731 flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Nov 20 2023
%Y A193731 Cf. A000012, A000079, A005053, A006234, A007483, A080420, A080421.
%Y A193731 Cf. A080422, A080423, A084938, A130129, A133494, A153881.
%Y A193731 Cf. A193722, A193730.
%K A193731 nonn,tabl
%O A193731 0,3
%A A193731 _Clark Kimberling_, Aug 04 2011