A158472 Triangle read by rows: n-th row is the expansion of the polynomial (x-F1)*(x-F2)*(x-F3)*...*(x-Fn).

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -7, 17, -17, 6, 1, -12, 52, -102, 91, -30, 1, -20, 148, -518, 907, -758, 240, 1, -33, 408, -2442, 7641, -12549, 10094, -3120, 1, -54, 1101, -11010, 58923, -173010, 273623, -215094, 65520

Offset: 0

Author

Gary W. Adamson, Mar 20 2009

Keywords

• tabl
• sign

Comments

Row sums of the unsigned triangle = A082480: (1, 2, 4, 12, 48, 288, 2592, ...).

Right border starting with row 1 (unsigned) = A003266: (1, 1, 2, 6, 30, 240, ...).

Examples

			First few rows of the unsigned triangle:
  1;
  1,  1;
  1,  2,    1;
  1,  4,    5,     2;
  1,  7,   17,    17,     6;
  1, 12,   52,   102,    91,     30;
  1, 20,  148,   518,   907,    758,    240;
  1, 33,  408,  2442,  7641,  12549,  10094,   3120;
  1, 54, 1101, 11010, 58923, 173010, 273623, 215094, 65520;
  ...
Example: row 5 is x^5 - 12x^4 + 52x^3 - 102x^2 + 91x - 30
= (x-1)*(x-1)*(x-2)*(x-3)*(x-5).
		

Links

• Alois P. Heinz, Rows n = 0..98, flattened

Crossrefs

Cf. A000045, A082480, A003266.

Programs

Maple

p:= proc(n) option remember; expand(`if`(n=0, 1,
      p(n-1)*(x-(<<0|1>, <1|1>>^n)[1, 2])))
    end:
T:= (n, k)-> coeff(p(n), x, n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Nov 06 2016

Mathematica

Array[Reverse@ CoefficientList[Times @@ Array[(x - Fibonacci@ #) &, #], x] &, 9, 0] // Flatten (* Michael De Vlieger, Apr 21 2019 *)

PARI

row(n) = Vec(prod(k=1, n, x-fibonacci(k)));
for (n=0, 10, print(row(n))); \\ Michel Marcus, Apr 22 2019

Extensions

One term corrected by Alois P. Heinz, Nov 06 2016