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A117413 Expansion of (1-x^2)/(1-2*x^2+4*x^3+x^4).

%I A117413 #15 Nov 22 2024 06:52:18
%S A117413 1,0,1,-4,1,-12,17,-24,81,-104,241,-508,817,-1876,3425,-6512,13537,
%T A117413 -24848,49697,-97332,185249,-368604,710129,-1380872,2709425,-5233656,
%U A117413 10232209,-19924140,38689617,-75543460,146843585,-285921248,557171393,-1083673376,2111184193,-4110111076
%N A117413 Expansion of (1-x^2)/(1-2*x^2+4*x^3+x^4).
%C A117413 Diagonal sums of number triangle A117411.
%H A117413 G. C. Greubel, <a href="/A117413/b117413.txt">Table of n, a(n) for n = 0..1000</a>
%H A117413 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,-4,-1).
%F A117413 a(n) = 2*a(n-2) - 4*a(n-3) - a(n-4).
%F A117413 a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2*k} C(n-k, k-j)*C(j, n-2*k)*(-4)^(n-2*k).
%F A117413 a(n) = (-)^n*A052992(2*n). - _R. J. Mathar_, Nov 22 2024
%t A117413 CoefficientList[Series[(1-x^2)/(1-2x^2+4x^3+x^4),{x,0,40}],x] (* or *) LinearRecurrence[{0,2,-4,-1},{1,0,1,-4},40] (* _Harvey P. Dale_, Jul 12 2017 *)
%o A117413 (Magma) I:=[1,0,1,-4]; [n le 4 select I[n] else 2*Self(n-1) -4*Self(n-2) -Self(n-3): n in [1..41]]; // _G. C. Greubel_, Sep 07 2022
%o A117413 (SageMath)
%o A117413 @CachedFunction
%o A117413 def a(n): # a = A117413
%o A117413     if(n<4): return (1, 0, 1, -4)[n]
%o A117413     else: return 2*a(n-2) - 4*a(n-3) - a(n-4)
%o A117413 [a(n) for n in (0..40)] # _G. C. Greubel_, Sep 07 2022
%Y A117413 Cf. A117411.
%K A117413 easy,sign
%O A117413 0,4
%A A117413 _Paul Barry_, Mar 14 2006