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A192877 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) given in Comments.

%I A192877 #15 Sep 08 2022 08:45:58
%S A192877 0,1,4,14,47,152,496,1601,5192,16786,54351,175836,569100,1841513,
%T A192877 5959484,19284934,62407951,201955408,653543000,2114907025,6843987040,
%U A192877 22147600586,71671151919,231932702004,750550018452,2428830833977
%N A192877 Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) given in Comments.
%C A192877 The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x + 1, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1.  See A192872.
%H A192877 G. C. Greubel, <a href="/A192877/b192877.txt">Table of n, a(n) for n = 0..1000</a>
%H A192877 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,6,-5,-6,4).
%F A192877 a(n) = 2*a(n-1) + 6*a(n-2) - 5*a(n-3) - 6*a(n-4) + 4*a(n-5).
%F A192877 G.f.: x*(1+2*x) / ( (1-x)*(1+x-x^2)*(1-2*x-4*x^2) ). - _R. J. Mathar_, May 06 2014
%t A192877 (See A192876.)
%t A192877 LinearRecurrence[{2,6,-5,-6,4}, {0,1,4,14,47}, 30] (* _G. C. Greubel_, Jan 08 2019 *)
%o A192877 (PARI) my(x='x+O('x^30)); concat([0], Vec(x*(1+2*x)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2)))) \\ _G. C. Greubel_, Jan 08 2019
%o A192877 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+2*x)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2)) )); // _G. C. Greubel_, Jan 08 2019
%o A192877 (Sage) (x*(1+2*x)/((1-x)*(1+x-x^2)*(1-2*x-4*x^2))).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 08 2019
%o A192877 (GAP) a:=[0,1,4,14,47];; for n in [6..30] do a[n]:=2*a[n-1]+6*a[n-2] -5*a[n-3]-6*a[n-4]+4*a[n-5]; od; a; # _G. C. Greubel_, Jan 08 2019
%Y A192877 Cf. A192872, A192876.
%K A192877 nonn
%O A192877 0,3
%A A192877 _Clark Kimberling_, Jul 11 2011