A193918 Mirror of the triangle A193917.

1, 1, 1, 3, 2, 1, 9, 6, 3, 2, 24, 15, 9, 5, 3, 64, 40, 24, 15, 8, 5, 168, 104, 64, 39, 24, 13, 8, 441, 273, 168, 104, 63, 39, 21, 13, 1155, 714, 441, 272, 168, 102, 63, 34, 21, 3025, 1870, 1155, 714, 440, 272, 165, 102, 55, 34, 7920, 4895, 3025, 1869, 1155

Offset: 0

Author

Clark Kimberling, Aug 09 2011

Keywords

• nonn
• tabl

Comments

A193918 is obtained by reversing the rows of the triangle A193917.

Here, we extend of the conjecture begun at A193917. Suppose n is an even positive integer and r(n+1,x) is the polynomial matched to row n+1 of A193918 as in the Mathematica program, where the first row is counted as row 0.

Conjecture: r(n+1,x) is the product of the following two polynomials whose coefficients are Fibonacci numbers:

linear factor: F(n+1)+x*F(n+2)

other factor: F(n+2)+F(n)*x^2+F(n-2)*x^4+...+F(2)*x^n.

Example, for n=4:

r(5,x)=64*x^5+40*x^4+24*x^3+15^x^2+8*x+5 factors as

8x+5 times 8x^4+3x^2+1.

Examples

			First six rows:
1
1....1
3....2....1
9....6....3....2
24...15...9....5....3
64...40...24...15...8...5
		

Crossrefs

Cf. A193917.

Programs

Mathematica

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
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}]]  (* A193917 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193918 *)

Formula

Write w(n,k) for the triangle at A193917. The triangle at A193918 is then given by w(n,n-k).