A192349 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

0, 1, 2, 14, 40, 180, 616, 2456, 8960, 34384, 128160, 485728, 1823360, 6882368, 25896064, 97614720, 367575040, 1384954112, 5216465408, 19651804672, 74025216000, 278859191296, 1050447030272, 3957059508224, 14906157629440, 56151566438400

Offset: 1

Author

Clark Kimberling, Jun 28 2011

Keywords

• nonn

Comments

To define the polynomials p(n,x), let d=sqrt(x+3); then p(n,x)=(1/2)((x+d)^n+(x-d)^n). These are similar to polynomials at A161516. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Examples

			The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=3+x+x^2 -> 4+2x
p(3,x)=9x+3x^2+x^3 -> 4+14x.
From these, we read
A192348=(1,0,3,4,...) and A192349=(0,1,2,14...)
		

Crossrefs

Cf. A192232, A192348.

Programs

Mathematica

(See A192348.)

Formula

Conjecture: a(n) = 2*a(n-1)+8*a(n-2)-4*a(n-3)-4*a(n-4). G.f.: x^2*(2*x^2+1) / (4*x^4+4*x^3-8*x^2-2*x+1). [Colin Barker, Jan 17 2013]