login
This site is supported by donations to The OEIS Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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: A115672 A079694 A068906 * A166347 A055300 A156256

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 16 03:17 EST 2019. Contains 319184 sequences. (Running on oeis4.)