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%I #15 Sep 08 2022 08:45:16
%S 1,1,1,1,0,-1,-3,-5,-7,-8,-7,-3,5,17,32,47,57,55,33,-16,-95,-199,-311,
%T -399,-416,-305,-11,499,1209,2024,2745,3061,2573,865,-2368,-7137,
%U -12943,-18577,-22015,-20512,-11007,9073,40593,81185,123712,155231,157165,107499,-14279,-219176
%N Expansion of (1-x^2)/(1-x-x^2+x^3+x^4).
%C Results from applying a Chebyshev transform after an inverse Catalan transform to 1/(1-x). The inverse Catalan transform maps g(x)->g(x(1-x)) while the Chebyshev transform maps h(x)->(1/(1+x^2))*h(x/(1+x^2)).
%H Muniru A Asiru, <a href="/A101496/b101496.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,-1)
%F a(n) = a(n-1) +a(n-2) -a(n-3) -a(n-4).
%F a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..floor((n-2k)/2)} C(n-k, k)*C(n-2k-j, j).
%t CoefficientList[Series[(1-x^2)/(1-x-x^2+x^3+x^4),{x,0,50}],x] (* or *) LinearRecurrence[{1,1,-1,-1},{1,1,1,1},50] (* _Harvey P. Dale_, Jun 05 2012 *)
%o (GAP) a:=[1,1,1,1];; for n in [5..50] do a[n]:=a[n-1]+a[n-2]-a[n-3]-a[n-4]; od; Print(a); # _Muniru A Asiru_, Mar 04 2019
%o (PARI) my(x='x+O('x^50)); Vec((1-x^2)/(1-x-x^2+x^3+x^4)) \\ _G. C. Greubel_, Mar 05 2019
%o (Magma) I:=[1,1,1,1]; [n le 4 select I[n] else Self(n-1) +Self(n-2) - Self(n-3) -Self(n-4): n in [1..50]]; // _G. C. Greubel_, Mar 05 2019
%o (Sage) ((1-x^2)/(1-x-x^2+x^3+x^4)).series(x, 50).coefficients(x, sparse=False) # _G. C. Greubel_, Mar 05 2019
%K easy,sign
%O 0,7
%A _Paul Barry_, Dec 04 2004
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