A078692 Triangle reads by rows: T(n,k) = coefficient of x^k in (x^3-2*x^2-2*x+1)^n.

1, -2, -2, 1, 1, -4, 0, 10, 0, -4, 1, 1, -6, 6, 19, -24, -24, 19, 6, -6, 1, 1, -8, 16, 20, -80, -8, 134, -8, -80, 20, 16, -8, 1, 1, -10, 30, 5, -160, 128, 330, -340, -340, 330, 128, -160, 5, 30, -10, 1, 1, -12, 48, -34, -240, 468, 399, -1416, -192, 2020, -192, -1416, 399, 468, -240, -34, 48, -12, 1

Offset: 1

Author

Mohammad K. Azarian, Feb 01 2003

Keywords

• sign
• tabf

Comments

Original name: Coefficients of polynomials in the denominator of the generating function f(x)=(x-x^2)/(x^3-2x^2-2x+1) for F(n)^2 (where F(n) is the Fibonacci sequence) and its successive derivatives starting with the highest power of x.

Examples

			Triangle begins:
  1, -2, -2, 1; # see A007598
  1, -4, 0, 10, 0, -4, 1;  # see A169630
  1, -6, 6, 19, -24, -24, 19, 6, -6, 1;
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11050 (rows 1..85 of triangle, flattened).

Crossrefs

Cf. A007598, A169630.

Programs

Mathematica

A078692row[n_] := Module[{x}, CoefficientList[(x^3 - 2*x^2 - 2*x + 1)^n, x]];
Array[A078692row, 6] (* Paolo Xausa, Jul 14 2025 *)

PARI

row(n) = Vec((x^3-2*x^2-2*x+1)^n); \\ Michel Marcus, Jul 11 2025

Formula

(d^(n)/d(x^n)) f(x), where f(x) = (x-x^2) / (x^3-2*x^2-2*x+1), for n=0, 1, 2, 3, ...

Extensions

Missing a(9) inserted and entry revised by Sean A. Irvine, Jul 11 2025