A137374 Triangular array of the coefficients of the Jacobsthal-Lucas polynomials ordered by descending powers, read by rows.

2, 1, 4, 1, 6, 1, 8, 8, 1, 20, 10, 1, 16, 36, 12, 1, 56, 56, 14, 1, 32, 128, 80, 16, 1, 144, 240, 108, 18, 1, 64, 400, 400, 140, 20, 1, 352, 880, 616, 176, 22, 1, 128, 1152, 1680, 896, 216, 24, 1, 832, 2912, 2912, 1248, 260, 26, 1, 256, 3136, 6272, 4704, 1680, 308, 28, 1

Offset: 0

Author

Roger L. Bagula, Apr 09 2008

Keywords

• nonn
• tabf

Comments

The even rows which start with 4, 8, 16 ... appear to be the absolute values of the Riordan array A128414. - Georg Fischer, Feb 25 2020

Examples

			The triangle starts:
    2;
    1;
    4,   1;
    6,   1;
    8,   8,   1;
   20,  10,   1;
   16,  36,  12,   1;
   56,  56,  14,   1;
   32, 128,  80,  16,  1;
  144, 240, 108,  18,  1;
   64, 400, 400, 140, 20, 1;
  352, 880, 616, 176, 22, 1;
  ...
		

Links

• Eric Weisstein's World of Mathematics, Jacobsthal-Lucas Polynomial.

Crossrefs

Row sums give A014551.

Cf. A034807.

Programs

Maple

b:= proc(n) option remember;
      `if`(n<2, 2-n, b(n-1)+2*expand(x*b(n-2)))
    end:
T:= n-> (p-> (d-> seq(coeff(p, x, d-i), i=0..d))(degree(p)))(b(n)):
seq(T(n), n=0..20);  # Alois P. Heinz, Feb 25 2020

Mathematica

f[0] = 2; f[1] = 1; f[n_] := 2 x f[n - 2] + f[n - 1];
Table[Reverse[CoefficientList[f[n], x]], {n, 0, 14}] // Flatten (* Jinyuan Wang, Feb 25 2020 *)

Formula

Let p(n, x) = 2*x*p(n-2, x) + p(n-1, x) with p(0, x) = 2 and p(1, x) = 1. The coefficients of the polynomial p(n, x), listed in reverse order, give row n. - Jinyuan Wang, Feb 25 2020

Extensions

Offset set to 0 by Peter Luschny, Feb 25 2020