A194001 Mirror of the triangle A194000.

1, 3, 2, 9, 5, 3, 24, 15, 8, 5, 64, 39, 24, 13, 8, 168, 104, 63, 39, 21, 13, 441, 272, 168, 102, 63, 34, 21, 1155, 714, 440, 272, 165, 102, 55, 34, 3025, 1869, 1155, 712, 440, 267, 165, 89, 55, 7920, 4895, 3024, 1869, 1152, 712, 432, 267, 144, 89, 20736

Offset: 0

Author

Clark Kimberling, Aug 11 2011

Keywords

• nonn
• tabl

Comments

A194001 is obtained by reversing the rows of the triangle A194000.

Here, we extend of the conjecture begun at A194000. Suppose n is an odd positive integer and r(n+1,x) is the polynomial matched to row n+1 of A194001 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+2)+x*F(n+3)

other: F(2)+F(4)*x^2+F(6)*x^4+...+F(n+1)*x^(n-1).

Example, for n=5:

r(6,x)=168*x^5+104*x^4+63*x^3+39^x^2+21*x+13 factors as

13+21x times 1+3x^2+8x^4.

Examples

			First six rows:
1
3....2
9....5....3
21...13...7....4
41...28...17...9....5
71...52...35...21...11...6
		

Crossrefs

Cf. A194000, A193918.

Programs

Mathematica

z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A194000 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A194001 *)

Formula

Write w(n,k) for the triangle at A194000. The triangle at A194001 is then given by w(n,n-k).