A193909 Mirror of the triangle A193908.

1, 1, 2, 3, 6, 8, 6, 12, 20, 24, 11, 22, 40, 64, 80, 19, 38, 72, 128, 208, 256, 32, 64, 124, 232, 416, 672, 832, 53, 106, 208, 400, 752, 1344, 2176, 2688, 87, 174, 344, 672, 1296, 2432, 4352, 7040, 8704, 142, 284, 564, 1112, 2176, 4192, 7872, 14080, 22784

Offset: 0

Author

Clark Kimberling, Aug 09 2011

Keywords

• nonn
• tabl

Comments

A193909 is obtained by reversing the rows of the triangle A193908.

Examples

			First six rows:
1
1....2
3....6....8
6....12...20...24
11...22...40...64....80
19...38...72...128...208...256
		

Crossrefs

Cf. A193908.

Programs

Mathematica

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 2]*x^(n - k), {k, 0, n}];
q[n_, x_] := 2 x*q[n - 1, x] + 1 ; q[0, x_] := 1;
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}]]  (* A193908 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193909 *)

Formula

Write w(n,k) for the triangle at A193908. The triangle at A193909 is then given by w(n,n-k).