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A192380 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192380 #12 Feb 19 2025 02:41:03
%S A192380 0,2,4,20,60,230,776,2792,9720,34410,120780,425788,1497716,5274190,
%T A192380 18562320,65348560,230024944,809742418,2850375060,10033806180,
%U A192380 35320352940,124333050422,437670231064,1540664252600,5423363437800,19091038878650,67203259647836
%N A192380 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192380 The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2).  For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%H A192380 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,6,-2,-1).
%F A192380 a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: 2*x^2 / (x^4+2*x^3-6*x^2-2*x+1). [_Colin Barker_, Dec 09 2012]
%e A192380 The first five polynomials p(n,x) and their reductions are as follows:
%e A192380 p(0,x)=1 -> 1
%e A192380 p(1,x)=2x -> 2x
%e A192380 p(2,x)=2+x+3x^2 -> 5+4x
%e A192380 p(3,x)=8x+4x^2+4x^3 -> 8+20x
%e A192380 p(4,x)=4+4x+21x^2+10x^3+5x^4 -> 45+60x.
%e A192380 From these, read
%e A192380 A192379=(1,0,5,8,45,...) and A192380=(0,2,4,20,60,...)
%t A192380 (See A192379.)
%Y A192380 Cf. A192232, A192379, A192381.
%K A192380 nonn,easy
%O A192380 1,2
%A A192380 _Clark Kimberling_, Jun 29 2011