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A192377 Coefficient of x in the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

%I A192377 #35 Aug 08 2025 09:08:48
%S A192377 0,2,4,20,68,262,968,3624,13512,50442,188236,702524,2621836,9784846,
%T A192377 36517520,136285264,508623504,1898208786,7084211604,26438637668,
%U A192377 98670339028,368242718486,1374300534872,5128959421048,19141537149272,71437189176090
%N A192377 Coefficient of x in the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.
%C A192377 The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2d), where d=sqrt(x+1). A192377=2*A192378. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%H A192377 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,6,2,-1).
%F A192377 From _Colin Barker_, Dec 09 2012: (Start)
%F A192377 a(n) = 2*a(n-1) + 6*a(n-2) + 2*a(n-3) - a(n-4).
%F A192377 G.f.: 2*x^2 / ((x+1)^2*(x^2-4*x+1)). (End)
%e A192377 The first five polynomials p(n,x) and their reductions are as follows:
%e A192377   p(0,x)=1 -> 1
%e A192377   p(1,x)=2x -> 2x
%e A192377   p(2,x)=4+x+3x^2 -> 7+4x
%e A192377   p(3,x)=16x+4x^2+4x^3 -> 16+20x
%e A192377   p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 73+68x.
%e A192377 From these, read (0,2,4,20,68,...)
%t A192377 (* See A192376. *)
%Y A192377 Cf. A192376, A192378.
%K A192377 nonn,easy
%O A192377 1,2
%A A192377 _Clark Kimberling_, Jun 29 2011