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

0,2

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

Links

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

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)

Example

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

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 *)

Prog

(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
(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

Crossrefs

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

Cf. A084938, A133494, A193722, A193729.

Sequence in context: A011019 A254979 A261157 * A295582 A364735 A192424

Adjacent sequences: A193725 A193726 A193727 * A193729 A193730 A193731

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 04 2011

Status

approved