A193924 Mirror of the triangle A193923.

1, 1, 1, 3, 2, 1, 8, 5, 3, 1, 21, 13, 8, 4, 1, 55, 34, 21, 12, 5, 1, 144, 89, 55, 33, 17, 6, 1, 377, 233, 144, 88, 50, 23, 7, 1, 987, 610, 377, 232, 138, 73, 30, 8, 1, 2584, 1597, 987, 609, 370, 211, 103, 38, 9, 1, 6765, 4181, 2584, 1596, 979, 581, 314, 141, 47

Offset: 0

Author

Clark Kimberling, Aug 09 2011

Keywords

• nonn
• tabl

Comments

A193924 is obtained by reversing the rows of the triangle A193923.

Examples

			First six rows:
1
1....1
3....2....1
8....5....3....1
21...13...8....4....1
55...34...21...12...5...1
		

Crossrefs

Cf. A193923.

Programs

Mathematica

p[n_, x_] := (x + 1)^n;
q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, 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}]]  (* A193923 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193924 *)

Formula

Write w(n,k) for the triangle at A193923. The triangle at A193924 is then given by w(n,n-k).