A137372 Triangle read by rows: coefficients of Fermat-Lucas polynomials.

2, 0, 3, -4, 0, 9, 0, -18, 0, 27, 8, 0, -72, 0, 81, 0, 60, 0, -270, 0, 243, -16, 0, 324, 0, -972, 0, 729, 0, -168, 0, 1512, 0, -3402, 0, 2187, 32, 0, -1152, 0, 6480, 0, -11664, 0, 6561, 0, 432, 0, -6480, 0, 26244, 0, -39366, 0, 19683, -64, 0, 3600, 0, -32400, 0, 102060, 0, -131220, 0, 59049

Offset: 0

Author

Roger L. Bagula, Apr 09 2008

Keywords

• tabl
• sign

Comments

The Fermat-Lucas polynomials F(n) are defined by the recurrence: F(0) = 2, F(1) = 3*y and F(n) = 3*y*F(n - 1) - 2*F(n - 2) for n > 1. - Andrew Howroyd, Aug 20 2018

Examples

			The first few polynomials are:
  2;
  3*y;
  -4 + 9*y^2;
  -18*y + 27*y^3;
  ...
Triangle begins:
    2;
    0, 3;
   -4, 0, 9;
    0, -18, 0, 27;
    8, 0, -72, 0, 81;
    0, 60, 0, -270, 0, 243;
  -16, 0,324, 0, -972, 0, 729;
    0, -168, 0, 1512, 0, -3402, 0, 2187;
   32, 0, -1152, 0, 6480, 0, -11664, 0, 6561;
    0, 432, 0, -6480, 0, 26244, 0, -39366, 0, 19683;
  -64, 0, 3600, 0, -32400, 0, 102060, 0, -131220, 0, 59049;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274
• Eric Weisstein's World of Mathematics, Fermat-Lucas Polynomial.

Crossrefs

Row sums are A000051.

Programs

Mathematica

<< Lucas`; Table[ExpandAll[Fermatf[n, x]], {n, 0, 10}]; a = Table[CoefficientList[Fermatf[n, x], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[Fermatf[n, x], x]], {n, 0, 10}]

PARI

T(n,k)=polcoef(polcoef((2 - 3*x*y)/(1 - 3*y*x + 2*x^2) + O(x*x^n), n, x), k, y);
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Aug 20 2018

Formula

G.f.: (2 - 3*x*y)/(1 - 3*y*x + 2*x^2). - Andrew Howroyd, Aug 21 2018

Extensions

Offset corrected and name clarified by Andrew Howroyd, Aug 20 2018