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 A193726 #27 Dec 03 2023 01:44:47
%S A193726 1,1,2,2,9,10,4,28,65,50,8,76,270,425,250,16,192,920,2200,2625,1250,
%T A193726 32,464,2800,9000,16250,15625,6250,64,1088,7920,32000,77500,112500,
%U A193726 90625,31250,128,2496,21280,103600,315000,612500,743750,515625,156250
%N A193726 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.
%C A193726 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%C A193726 Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (2,3,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 05 2011
%H A193726 G. C. Greubel, <a href="/A193726/b193726.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A193726 T(n,k) = 5*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - _Philippe Deléham_, Oct 05 2011
%F A193726 G.f.: (1-x-3*x*y)/(1-2*x-5*x*y). - _R. J. Mathar_, Aug 11 2015
%F A193726 From _G. C. Greubel_, Dec 02 2023: (Start)
%F A193726 T(n, 0) = A011782(n).
%F A193726 T(n, n) = A020699(n).
%F A193726 T(n, n-1) = A081040(n-1).
%F A193726 Sum_{k=0..n} T(n, k) = A169634(n-1) + (4/7)*[n=0].
%F A193726 Sum_{k=0..n} (-1)^k * T(n, k) = (-1)^n*A133494(n) = -A141413(n+1).
%F A193726 Sum_{k=0..floor(n/2)} T(n-k, k) = A002532(n) + 2*A002532(n-1) + (3/5)*[n=0].
%F A193726 Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A045873(n) - 2*A045873(n-1) + (3/5)*[n=0]. (End)
%e A193726 First six rows:
%e A193726 1;
%e A193726 1, 2;
%e A193726 2, 9, 10;
%e A193726 4, 28, 65, 50;
%e A193726 8, 76, 270, 425, 250;
%e A193726 16, 192, 920, 2200, 2625, 1250;
%t A193726 (* First program *)
%t A193726 z = 8; a = 1; b = 2; c = 1; d = 2;
%t A193726 p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
%t A193726 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193726 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193726 g[n_] := CoefficientList[w[n, x], {x}]
%t A193726 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193726 Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193726 *)
%t A193726 TableForm[Table[g[n], {n, -1, z}]]
%t A193726 Flatten[Table[g[n], {n, -1, z}]] (* A193727 *)
%t A193726 (* Second program *)
%t A193726 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, k+1, 2*T[n-1, k] + 5*T[n -1, k-1]]];
%t A193726 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 02 2023 *)
%o A193726 (Magma)
%o A193726 function T(n, k) // T = A193726
%o A193726 if k lt 0 or k gt n then return 0;
%o A193726 elif n lt 2 then return k+1;
%o A193726 else return 2*T(n-1, k) + 5*T(n-1, k-1);
%o A193726 end if;
%o A193726 end function;
%o A193726 [T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 02 2023
%o A193726 (SageMath)
%o A193726 def T(n, k): # T = A193726
%o A193726 if (k<0 or k>n): return 0
%o A193726 elif (n<2): return k+1
%o A193726 else: return 2*T(n-1, k) + 5*T(n-1, k-1)
%o A193726 flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Dec 02 2023
%Y A193726 Cf. A002532, A011782, A020699, A045873, A081040, A084938.
%Y A193726 Cf. A133494, A141413, A169634, A193722, A193727.
%K A193726 nonn,tabl
%O A193726 0,3
%A A193726 _Clark Kimberling_, Aug 04 2011