A193729 Mirror of the triangle A193728.

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

Offset: 0

Author

Clark Kimberling, Aug 04 2011

Keywords

• nonn
• tabl

Comments

T(n, k) is obtained by reversing the rows of the triangle A193728.

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

Examples

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

Links

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

Crossrefs

Cf. A000420, A033999, A081038, A081294, A084938.

Cf. A108981, A133494, A193722, A193728, A247560.

Programs

Magma

function T(n, k) // T = A193729
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return k+1;
  else return 3*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 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, k+1, 3*T[n-1,k] + 4*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 = A193729
    if (k<0 or k>n): return 0
    elif (n<2): return k+1
    else: return 3*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 28 2023

Formula

Let w(n,k) be the triangle of A193728, then the triangle in this sequence is given by w(n,n-k).

T(n,k) = 4*T(n-1,k-1) + 3*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-2*x*y)/(1-3*x-4*x*y). - R. J. Mathar, Aug 11 2015

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

T(n, 0) = A133494(n).

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

T(n, n) = A081294(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) = A033999(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*[n=0] + A108981(n-1).

Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = (1/2)*[n=0] + A247560(n-1).

(End)