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%I #52 Feb 28 2024 15:59:20
%S 1,-1,-1,7,-17,23,-1,-89,271,-457,287,967,-4049,8279,-8641,-7193,
%T 56143,-139657,194399,-24569,-703889,2209943,-3814273,2603047,7447951,
%U -32756041,68476319,-74404793,-50690897,449691863,-1146312001,1640168551,-335257649,-5554901257
%N Expansion of (1 + 2*x)/(1 + 3*x + 4*x^2).
%C For positive n, a(n) equals 2^n times the permanent of the (2n) X (2n) tridiagonal matrix with 1/sqrt(2)'s along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - _John M. Campbell_, Jul 08 2011
%C For n > 3, equals -1 times the determinant of the (n-2) X (n-2) matrix with 2^2's along the superdiagonal, 3^2's along the main diagonal, 4^2's along the subdiagonal, etc., and 0's everywhere else. - _John M. Campbell_, Dec 01 2011
%H Vincenzo Librandi, <a href="/A087168/b087168.txt">Table of n, a(n) for n = 0..1000</a>
%H Taras Goy and Mark Shattuck, <a href="https://doi.org/10.2478/amsil-2023-0027">Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries</a>, Ann. Math. Silesianae (2023). See p. 14.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-4).
%F G.f.: (1+2*x)/(1+3*x+4*x^2).
%F a(n) = -3*a(n-1) - 4*a(n-2); a(0)=1, a(1)=-1.
%F a(n) = Sum_{k=0..n} C(n+k,2*k)*(-2)^(n-k).
%F a(n) = -a(-1-n) * 2^(2*n+1) = A001607(2*n + 1) for all n in Z. - _Michael Somos_, Sep 19 2014
%F a(n) = (-2)^(n-1)*(2*ChebyshevU(n-2, 3/4) - ChebyshevU(n-1, 3/4)). - _G. C. Greubel_, Jun 09 2022
%e G.f. = 1 - x - x^2 + 7*x^3 - 17*x^4 + 23*x^5 - x^6 - 89*x^7 + 271*x^8 + ...
%t CoefficientList[Series[(1+2x)/(1+3x+4x^2), {x, 0, 30}], x]
%t Table[-Det[Array[Sum[KroneckerDelta[#1, #2+q]*(q+3)^2, {q, -1, n-2}] &, {n-2, n-2}]], {n, 4, 50}] (* _John M. Campbell_, Dec 01 2011 *)
%t LinearRecurrence[{-3,-4},{1,-1},40] (* _Harvey P. Dale_, Apr 23 2014 *)
%o (Magma)
%o A087168:= func< n | &+[ Binomial(n+k, 2*k)*(-2)^(n-k): k in [0..n] ] >;
%o [A087168(n): n in [0..35]];
%o (PARI) {a(n) = real( (-1 - quadgen(-7))^n )}; /* _Michael Somos_, Sep 19 2014 */
%o (SageMath)
%o def A087168(n): return (-2)^(n-1)*(2*chebyshev_U(n-2, 3/4) -chebyshev_U(n-1, 3/4))
%o [A087168(n) for n in (0..50)] # _G. C. Greubel_, Jun 09 2022
%Y Cf. A001607, A049072.
%K easy,sign
%O 0,4
%A Mario Catalani (mario.catalani(AT)unito.it), Aug 22 2003
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