A162514 Triangle of coefficients of polynomials defined by the Binet form P(n,x) = U^n + L^n, where U = (x + d)/2, L = (x - d)/2, d = (4 + x^2)^(1/2). Decreasing powers of x.

2, 1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 4, 0, 2, 1, 0, 5, 0, 5, 0, 1, 0, 6, 0, 9, 0, 2, 1, 0, 7, 0, 14, 0, 7, 0, 1, 0, 8, 0, 20, 0, 16, 0, 2, 1, 0, 9, 0, 27, 0, 30, 0, 9, 0, 1, 0, 10, 0, 35, 0, 50, 0, 25, 0, 2, 1, 0, 11, 0, 44, 0, 77, 0, 55, 0, 11, 0, 1, 0, 12, 0, 54, 0, 112, 0, 105, 0, 36, 0, 2, 1, 0, 13, 0

Offset

0,1

Comments

For a signed version of this triangle corresponding to the row reversed version of the triangle A127672 see A244422. - Wolfdieter Lang, Aug 07 2014

The row reversed triangle is A114525. - Paolo Bonzini, Jun 23 2016

Links

G. C. Greubel, Rows n=0..100 of triangle, flattened

Formula

P(n,x) = x*P(n-1,x) + P(n-2,x) for n >= 2, P(0,x) = 2, P(1,x) = x.

From Wolfdieter Lang, Aug 07 2014: (Start)

T(n,m) = [x^(n-m)] P(n,x), m = 0, 1, ..., n and n >= 0.

G.f. of polynomials P(n,x): (2 - x*z)/(1 - x*z - z^2).

G.f. of row polynomials R(n,x) = Sum_{m=0..n} T(n,m)*x^m: (2 - z)/(1 - z - (x*z)^2) (rows for P(n,x) reversed).

(End)

For n > 0, T(n,2*m+1) = 0, T(n,2*m) = A034807(n,m). - Paolo Bonzini, Jun 23 2016

Example

Triangle begins
   2;  == 2
   1, 0;  == x + 0
   1, 0,  2;  == x^2 + 2
   1, 0,  3, 0;  == x^3 + 3*x + 0
   1, 0,  4, 0,  2;
   1, 0,  5, 0,  5, 0;
   1, 0,  6, 0,  9, 0,  2;
   1, 0,  7, 0, 14, 0,  7, 0;
   1, 0,  8, 0, 20, 0, 16, 0,  2;
   1, 0,  9, 0, 27, 0, 30, 0,  9, 0;
   1, 0, 10, 0, 35, 0, 50, 0, 25, 0, 2;
   ...
From Wolfdieter Lang, Aug 07 2014: (Start)
The row polynomials R(n, x) are:
  R(0, x) = 2, R(1, x) = 1 =   x*P(1,1/x),  R(2, x) = 1 + 2*x^2 = x^2*P(2,1/x), R(3, x) = 1 + 3*x^2 = x^3*P(3,1/x), ...
(End)

Mathematica

Table[Reverse[CoefficientList[LucasL[n, x], x]], {n, 0, 12}]//Flatten  (* G. C. Greubel, Nov 05 2018 *)

Prog

(PARI)
P(n)=
{
    local(U, L, d, r, x);
    if ( n<0, return(0) );
    x = 'x+O('x^(n+1));
    d=(4 + x^2)^(1/2);
    U=(x+d)/2;  L=(x-d)/2;
    r = U^n+L^n;
    r = truncate(r);
    return( r );
}
for (n=0, 10, print(Vec(P(n))) ); /* show triangle */
/* Joerg Arndt, Jul 24 2011 */

Crossrefs

Cf. A000032, A114525, A162515, A162516, A162517.

Sequence in context: A079694 A339895 A068906 * A364039 A376362 A166347

Adjacent sequences: A162511 A162512 A162513 * A162515 A162516 A162517

Keyword

nonn,tabl

Author

Clark Kimberling, Jul 05 2009

Extensions

Name clarified by Wolfdieter Lang, Aug 07 2014

Status

approved