%I #56 Jul 01 2023 15:04:40
%S 4,8,24,80,272,928,3168,10816,36928,126080,430464,1469696,5017856,
%T 17132032,58492416,199705600,681837568,2327939072,7948081152,
%U 27136446464,92649623552,316325601280,1080003158016,3687361429504,12589439401984,42983034748928
%N Non-vanishing traces of the powers of the adjacency matrix for the simple Lie algebra B_4: 2 * ((2 + sqrt(2))^n + (2 - sqrt(2))^n).
%C a(n) is the trace of the 2n-th power of the adjacency matrix M for the simple Lie algebra B_4, given in the Damianou link. M = Matrix[row 1; row 2; row 3; row 4] = Matrix[0,1,0,0; 1,0,1,0; 0,1,0,2; 0,0,1,0]. Equivalently, the trace tr(M^(2k)) is the sum of the 2n-th powers of the eigenvalues of M. The eigenvalues are the zeros of the characteristic polynomial of M, which is det(xI - M) = x^4 - 4x^2 + 2 = A127672(4,x), and are (+-) sqrt((2 + sqrt(2)) and (+-) sqrt((2 - sqrt(2)), or the four unique values generated by 2*cos((2n+1)*Pi/8). Compare with A025192 for B_3. The odd power traces vanish.
%C -log(1 - 4*x^2 + 2*x^4) = 8*x^2/2 + 24*x^4/4 + 80*x^6/6 + ... = Sum_{n>0} tr(M^k) x^k / k = Sum_{n>0} a(n) x^(2k) / 2k gives an aerated version of the sequence a(n), excluding a(0), and exp(-log(1 - 4*x + 2*x^2)) = 1 / (1 - 4*x + 2*x^2) is the e.g.f. for A007070.
%C As in A025192, the cycle index partition polynomials P_k(x[1],...,x[k]) of A036039 evaluated with the negated power sums, the aerated a(n), are P_2(0,-a(1)) = P_2(0,-8) = -8, P_4(0,-a(1),0,-a(2)) = P_4(0,-8,0,-24) = 48, and all other P_k(0,-a(1),0,-a(2),0,...) = 0 since 1 - 4*x^2 + 2*x^4 = 1 - 8*x^2/2! + 48*x^4/4! = det(I - x M) = exp(-Sum_{k>0} tr(M^k) x^k / k) = exp[P.(-tr(M),-tr(M^2),...)x] = exp[P.(0,-a(1),0,-a(2),...)x].
%C Because of the inverse relation between the Faber polynomials F_n(b1,b2,...,bn) of A263916 and the cycle index polynomials, F_n(0,-4,0,2,0,0,0,...) = tr(M^n) gives aerated a(n), excluding a(0). E.g., F_2(0,-4) = -2 * -4 = 8, F_4(0,-4,0,2) = -4 * 2 + 2 * (-4)^2 = 24, and F_6(0,-4,0,2,0,0) = -2*(-4)^3 + 6*(-4)*2 = 80.
%H G. C. Greubel, <a href="/A265185/b265185.txt">Table of n, a(n) for n = 0..1000</a>
%H P. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2011-2014.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4, -2).
%F a(n) = 2 * ((2 + sqrt(2))^n + (2 - sqrt(2))^n) = Sum_{k=0..3} 2^(2n) (cos((2k+1)*Pi/8))^(2n) = 2*2^(2n) (cos(Pi/8)^(2n) + cos(3*Pi/8)^(2n)) = 2 Sum_{k=0..1} (exp(i(2k+1)*Pi/8) + exp(-i*(2k+1)*Pi/8))^(2n).
%F E.g.f.: 2 * e^(2*x) * (e^(sqrt(2)*x) + e^(-sqrt(2)*x)) = 4*e^(2*x)*cosh(sqrt(2)*x) = 2*(exp(4*x*cos(Pi/8)^2) + exp(4*x cos(3*Pi/8)^2) ).
%F a(n) = 4*A006012(n) = 8*A007052(n-1) = 2*A056236(n).
%F G.f.: (4-8*x)/(1-4*x+2*x^2). - _Robert Israel_, Dec 07 2015
%F Note the preceding o.g.f. is four times that of A006012 and the denominator is y^4 * A127672(4,1/y) with y = sqrt(x). Compare this with those of A025192 and A189315. - _Tom Copeland_, Dec 08 2015
%t 4 LinearRecurrence[{4, -2}, {1, 2}, 30] (* _Vincenzo Librandi_, Dec 06 2015 and slightly modified by _Robert G. Wilson v_, Feb 13 2018 *)
%o (Magma) [Floor(2 * ((2 + Sqrt(2))^n + (2 - Sqrt(2))^n)): n in [0..30]]; // _Vincenzo Librandi_, Dec 06 2015
%o (PARI) x='x+O('x^30); Vec((4-8*x)/(1-4*x+2*x^2)) \\ _G. C. Greubel_, Feb 12 2018
%Y Cf. A006012, A007052, A007070, A025192, A036039, A056236, A127672, A189315, A263916.
%K nonn,easy
%O 0,1
%A _Tom Copeland_, Dec 04 2015
%E More terms from _Vincenzo Librandi_, Dec 06 2015