A193734 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.

1, 1, 2, 1, 6, 8, 1, 10, 32, 32, 1, 14, 72, 160, 128, 1, 18, 128, 448, 768, 512, 1, 22, 200, 960, 2560, 3584, 2048, 1, 26, 288, 1760, 6400, 13824, 16384, 8192, 1, 30, 392, 2912, 13440, 39424, 71680, 73728, 32768, 1, 34, 512, 4480, 25088, 93184, 229376, 360448, 327680, 131072

Offset

0,3

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 (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

Links

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

Formula

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

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

From G. C. Greubel, Nov 19 2023: (Start)

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

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

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

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

Example

First six rows:
  1;
  1,  2;
  1,  6,   8;
  1, 10,  32,  32;
  1, 14,  72, 160, 128;
  1, 18, 128, 448, 768, 512;

Mathematica

(* First program *)
z = 8; a = 2; b = 1; c = 1; d = 2;
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}]] (* A193734 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]      (* A193735 *)
(* Second program *)
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]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//TableForm (* G. C. Greubel, Nov 19 2023 *)

Prog

(Magma)
function T(n, k) // T = A193734
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return T(n-1, k) + 4*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2023
(SageMath)
def T(n, k): # T = A193734
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return T(n-1, k) +4*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 19 2023

Crossrefs

Cf. A084938, A193722, A193735.

Cf. A005053, A026581, A081294, A133494.

Sequence in context: A319897 A350009 A350010 * A318390 A319511 A110608

Adjacent sequences: A193731 A193732 A193733 * A193735 A193736 A193737

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 04 2011

Status

approved