A193728 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (x+2)^n and q(n,x) = (2*x+1)^n.

1, 2, 1, 8, 10, 3, 32, 64, 42, 9, 128, 352, 360, 162, 27, 512, 1792, 2496, 1728, 594, 81, 2048, 8704, 15360, 14400, 7560, 2106, 243, 8192, 40960, 87552, 103680, 73440, 31104, 7290, 729, 32768, 188416, 473088, 677376, 604800, 344736, 122472, 24786, 2187

Offset: 0

Author

Clark Kimberling, Aug 04 2011

Keywords

• nonn
• tabl

Comments

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

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

Examples

			First six rows:
    1;
    2,    1;
    8,   10,    3;
   32,   64,   42,    9;
  128,  352,  360,  162,  27;
  512, 1792, 2496, 1728, 594, 81;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000012, A000420, A060816, A081038, A081294.

Cf. A084938, A133494, A193722, A193729.

Programs

Magma

function T(n, k) // T = A193728
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 4*T(n-1, k) + 3*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 28 2023

Mathematica

(* First program *)
z = 8; a = 1; b = 2; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193728 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193729 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 4*T[n-1,k] + 3*T[n-1,k-1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 28 2023 *)

SageMath

def T(n, k): # T = A193728
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 4*T(n-1, k) + 3*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 28 2023

Formula

T(n,k) = 3*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

G.f.: (1-2*x-2*x*y)/(1-4*x-3*x*y). - R. J. Mathar, Aug 11 2015

From G. C. Greubel, Nov 28 2023: (Start)

T(n, n-k) = A193729(n, k).

T(n, 0) = A081294(n).

T(n, n-1) = 2*A081038(n-1).

T(n, n) = A133494(n).

Sum_{k=0..n} T(n, k) = (1/7)*(4*[n=0] + 3*A000420(n)).

Sum_{k=0..n} (-1)^k * T(n, k) = A000012(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = (5*b(n) + 4*b(n-1))/14 + (2/3)*[n=0].

Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A060816(n),

where b(n) = (2 + sqrt(7))^n + (2 - sqrt(7))^n. (End)