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

0,2

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.

Links

Table of n, a(n) for n=0..55.

Formula

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

Example

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

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 *)

Crossrefs

Cf. A194000, A193918.

Sequence in context: A346105 A188926 A193980 * A178230 A268824 A306470

Adjacent sequences: A193998 A193999 A194000 * A194002 A194003 A194004

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 11 2011

Status

approved