A265185 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).

4, 8, 24, 80, 272, 928, 3168, 10816, 36928, 126080, 430464, 1469696, 5017856, 17132032, 58492416, 199705600, 681837568, 2327939072, 7948081152, 27136446464, 92649623552, 316325601280, 1080003158016, 3687361429504, 12589439401984, 42983034748928

Offset: 0

Author

Tom Copeland, Dec 04 2015

Keywords

• nonn
• easy

Comments

a(n) is the trace of the 2*n-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^(2*k)) is the sum of the 2*n-th powers of the eigenvalues of M. The eigenvalues are the zeros of the characteristic polynomial of M, which is det(x*I - M) = x^4 - 4*x^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((2*n+1)*Pi/8). Compare with A025192 for B_3. The odd power traces vanish.

-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.

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].

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2011-2014.
• Index entries for linear recurrences with constant coefficients, signature (4,-2).

Crossrefs

Cf. A006012, A007052, A007070, A025192, A036039, A056236, A127672, A189315, A263916.

Programs

Magma

[Floor(2 * ((2 + Sqrt(2))^n + (2 - Sqrt(2))^n)): n in [0..30]]; // Vincenzo Librandi, Dec 06 2015

Mathematica

4 LinearRecurrence[{4, -2}, {1, 2}, 30] (* Vincenzo Librandi, Dec 06 2015 and slightly modified by Robert G. Wilson v, Feb 13 2018 *)

PARI

my(x='x+O('x^30)); Vec((4-8*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Feb 12 2018

Formula

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).

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) ).

a(n) = 4*A006012(n) = 8*A007052(n-1) = 2*A056236(n).

G.f.: (4-8*x)/(1-4*x+2*x^2). - Robert Israel, Dec 07 2015

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

Extensions

More terms from Vincenzo Librandi, Dec 06 2015