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%I #16 Sep 08 2022 08:45:28
%S 0,0,0,1,0,3,-4,5,-24,19,-76,133,-208,627,-852,2181,-4232,7443,-18012,
%T 30533,-66880,133875,-250724,547013,-1020152,2108435,-4245612,8217861,
%U -17089968,33202291,-67158900,135095301,-265925992,541112339,-1069523580,2146659781,-4309316128,8553624307
%N Expansion of g.f.: x^4/((1+2*x) * (1-2*x+x^2+2*x^3)).
%H G. C. Greubel, <a href="/A123957/b123957.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,-4,-4).
%F a(n)= 3*a(n-2) -4*a(n-3) -4*a(n-4).
%F a(n)= ((-2)^n -9*A077942(n) +20*A077942(n-1) -17*A077942(n-2))/32, n>3.
%p seq(coeff(series(x^4/((1+2*x)*(1-2*x+x^2+2*x^3)), x, n+1), x, n), n = 1..40); # _G. C. Greubel_, Aug 05 2019
%t M = {{0,1,0,0}, {0,0,1,0}, {0,0,0,1}, {-4,-4,3,0}}; v[1] = {0,0,0,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n, 40}]
%t Rest@CoefficientList[Series[x^4/((1+2*x)*(1-2*x+x^2+2*x^3)),{x,0,40}],x] (* or *) LinearRecurrence[{0,3,-4,-4},{0,0,0,1},40] (* _Harvey P. Dale_, Dec 27 2015 *)
%o (PARI) my(x='x+O('x^40)); concat([0,0,0], Vec(x^4/((1+2*x)*(1-2*x+x^2+ 2*x^3)))) \\ _G. C. Greubel_, Aug 05 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0,0,0] cat Coefficients(R!( x^4/((1+2*x)*(1-2*x+x^2+2*x^3)) )); // _G. C. Greubel_, Aug 05 2019
%o (Sage) a=(x^4/((1+2*x) * (1-2*x+x^2+2*x^3))).series(x, 40).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Aug 05 2019
%o (GAP) a:=[0, 0, 0, 1];; for n in [5..40] do a[n]:=3*a[n-2]-4*a[n-3] -4*a[n-4]; od; a; # _G. C. Greubel_, Aug 05 2019
%K sign
%O 1,6
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 27 2006
%E Definition replaced with generating function. - the Assoc. Eds of the OEIS, Mar 28 2010
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