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

0,1

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

Links

Table of n, a(n) for n=0..63.

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

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

Example

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;
  ...

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 *)

Crossrefs

Row sums give A014551.

Cf. A034807.

Sequence in context: A339420 A317494 A317505 * A131516 A088140 A130758

Adjacent sequences: A137371 A137372 A137373 * A137375 A137376 A137377

Keyword

nonn,tabf

Author

Roger L. Bagula, Apr 09 2008

Extensions

Offset set to 0 by Peter Luschny, Feb 25 2020

Status

approved