A193725 Mirror of the triangle A193724.

1, 1, 1, 3, 5, 2, 9, 21, 16, 4, 27, 81, 90, 44, 8, 81, 297, 432, 312, 112, 16, 243, 1053, 1890, 1800, 960, 272, 32, 729, 3645, 7776, 9180, 6480, 2736, 640, 64, 2187, 12393, 30618, 43092, 37800, 21168, 7392, 1472, 128, 6561, 41553, 116640, 190512, 199584, 139104, 64512, 19200, 3328, 256

Offset: 0

Author

Clark Kimberling, Aug 04 2011

Keywords

• nonn
• tabl

Comments

A193725 is obtained by reversing the rows of the triangle A193724.

Triangle T(n,k), read by rows, given by [1,2,0,0,0,0,...] DELTA [1,1,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 04 2011

Examples

			First six rows:
   1;
   1,   1;
   3,   5,   2;
   9,  21,  16,   4;
  27,  81,  90,  44,   8;
  81, 297, 432, 312, 112, 16;
		

Crossrefs

Cf. A084938, A193724.

Programs

Mathematica

z = 8; a = 1; b = 2; c = 1; 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}]] (* A193724 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193725 *)

Formula

Write w(n,k) for the triangle at A193724. The triangle at A193725 is then given by w(n,n-k).

T(n,k) = 2*T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,0)=T(1,1)=1. - Philippe Deléham, Oct 05 2011

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