A162514 - OEIS

%I #27 Nov 06 2018 06:45:48

%S 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,

%T 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,

%U 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

%N 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.

%C For a signed version of this triangle corresponding to the row reversed version of the triangle A127672 see A244422. - _Wolfdieter Lang_, Aug 07 2014

%C The row reversed triangle is A114525. - _Paolo Bonzini_, Jun 23 2016

%H G. C. Greubel, <a href="/A162514/b162514.txt">Rows n=0..100 of triangle, flattened</a>

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

%F From _Wolfdieter Lang_, Aug 07 2014: (Start)

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

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

%F 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).

%F (End)

%F For n > 0, T(n,2*m+1) = 0, T(n,2*m) = A034807(n,m). - _Paolo Bonzini_, Jun 23 2016

%e Triangle begins

%e    2;  == 2

%e    1, 0;  == x + 0

%e    1, 0,  2;  == x^2 + 2

%e    1, 0,  3, 0;  == x^3 + 3*x + 0

%e    1, 0,  4, 0,  2;

%e    1, 0,  5, 0,  5, 0;

%e    1, 0,  6, 0,  9, 0,  2;

%e    1, 0,  7, 0, 14, 0,  7, 0;

%e    1, 0,  8, 0, 20, 0, 16, 0,  2;

%e    1, 0,  9, 0, 27, 0, 30, 0,  9, 0;

%e    1, 0, 10, 0, 35, 0, 50, 0, 25, 0, 2;

%e    ...

%e From _Wolfdieter Lang_, Aug 07 2014: (Start)

%e The row polynomials R(n, x) are:

%e   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), ...

%e (End)

%t Table[Reverse[CoefficientList[LucasL[n, x], x]], {n, 0, 12}]//Flatten  (* _G. C. Greubel_, Nov 05 2018 *)

%o (PARI)

%o P(n)=

%o {

%o     local(U, L, d, r, x);

%o     if ( n<0, return(0) );

%o     x = 'x+O('x^(n+1));

%o     d=(4 + x^2)^(1/2);

%o     U=(x+d)/2;  L=(x-d)/2;

%o     r = U^n+L^n;

%o     r = truncate(r);

%o     return( r );

%o }

%o for (n=0, 10, print(Vec(P(n))) ); /* show triangle */

%o /* _Joerg Arndt_, Jul 24 2011 */

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

%K nonn,tabl

%O 0,1

%A _Clark Kimberling_, Jul 05 2009

%E Name clarified by _Wolfdieter Lang_, Aug 07 2014