%I #26 Nov 20 2023 00:02:58

%S 1,1,2,1,6,8,1,10,32,32,1,14,72,160,128,1,18,128,448,768,512,1,22,200,

%T 960,2560,3584,2048,1,26,288,1760,6400,13824,16384,8192,1,30,392,2912,

%U 13440,39424,71680,73728,32768,1,34,512,4480,25088,93184,229376,360448,327680,131072

%N Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=(x+2)^n.

%C See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

%C Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,...) DELTA (2,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Oct 05 2011

%H G. C. Greubel, <a href="/A193734/b193734.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n,k) = 4*T(n-1,k-1) + T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - _Philippe Deléham_, Oct 05 2011

%F G.f.: (1 - 2*x*y)/(1 - x - 4*x*y). - _R. J. Mathar_, Aug 11 2015

%F From _G. C. Greubel_, Nov 19 2023: (Start)

%F T(n, n) = A081294(n).

%F Sum_{k=0..n} T(n, k) = A005053(n).

%F Sum_{k=0..n} (-1)^k * T(n, k) = (-1)^n * A133494(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A026581(n-1) + (1/2)*[n=0]. (End)

%e First six rows:

%e 1;

%e 1, 2;

%e 1, 6, 8;

%e 1, 10, 32, 32;

%e 1, 14, 72, 160, 128;

%e 1, 18, 128, 448, 768, 512;

%t (* First program *)

%t z = 8; a = 2; b = 1; c = 1; d = 2;

%t p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n

%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

%t g[n_] := CoefficientList[w[n, x], {x}]

%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]

%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193734 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]] (* A193735 *)

%t (* Second program *)

%t T[n_, k_]:= T[n,k]= If[k<0 || k>n, 0, If[n<2, k+1, T[n-1,k] +4*T[n-1,k-1]]];

%t Table[T[n,k], {n,0,12}, {k,0,n}]//TableForm (* _G. C. Greubel_, Nov 19 2023 *)

%o (Magma)

%o function T(n, k) // T = A193734

%o if k lt 0 or k gt n then return 0;

%o elif n lt 2 then return k+1;

%o else return T(n-1, k) + 4*T(n-1, k-1);

%o end if;

%o end function;

%o [T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 19 2023

%o (SageMath)

%o def T(n, k): # T = A193734

%o if (k<0 or k>n): return 0

%o elif (n<2): return k+1

%o else: return T(n-1, k) +4*T(n-1, k-1)

%o flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Nov 19 2023

%Y Cf. A084938, A193722, A193735.

%Y Cf. A005053, A026581, A081294, A133494.

%K nonn,tabl

%O 0,3

%A _Clark Kimberling_, Aug 04 2011