A188067 Expansion of x^2*(x^3+2*x^2+x+1)/((x-1)*(x+1))^4.

0, 0, 1, 1, 6, 5, 18, 14, 40, 30, 75, 55, 126, 91, 196, 140, 288, 204, 405, 285, 550, 385, 726, 506, 936, 650, 1183, 819, 1470, 1015, 1800, 1240, 2176, 1496, 2601, 1785, 3078, 2109, 3610, 2470, 4200, 2870, 4851, 3311, 5566, 3795, 6348, 4324, 7200, 4900, 8125

Offset

0,5

Comments

(Start) Let {U_N(x)}, N=0,1,2,..., be Chebyshev polynomials of the second kind, defined by U_0(x)=1, U_1(x)=2*x, U_N(x)=2*x*U_(N-1)(x)-U_(N-2)(x). Let n>2 and q_n=floor(n/2). Let A^<n>={A_(n,0),A_(n,1),...,A_(n,q_n-1)} be the n-th set of unit-primitive matrices (see [Jeffery]). We form a matrix containing the ordered spectra of the unit-primitives from A^<n> as follows. Define the column vectors S(A_(n,r))=Specrum(A_(n,r))=(U_r(x_0),U_r(x_1),...,U_r(x_(q_n-1)))^T, with x_r=cos((2*r+1)*Pi/n), r=0,1,...,q_n-1, where T denotes the transpose. Let S^<n> be the q_n X q_n matrix formed by combining the S(A_(n,r)) such that

S^<n>=[S(A_(n,0)),S(A_(n,1)),...,S(A_(n,q_n-1))]=

(1, U_1(x_0),...,U_(q_n-1)(x_0));

(1, U_1(x_1),...,U_(q_n-1)(x_1));

... (etc.)

(1, U_1(x_(q_n-1)),...,U_(q_n-1)(x_(q_n-1))),

so column r contains the (ordered) spectrum of A_(n,r). Then for our sequence, let a(0)=a(1)=0 and a(2)=a(3)=1, and, for n>3, we have

a(n)=Trace((S^<n>)^T*S^<n>). (End)

Links

Table of n, a(n) for n=0..50.

L. E. Jeffery, Unit-primitive matrices

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Formula

G.f.: x^2*(x^3+2*x^2+x+1)/((x-1)*(x+1))^4.

a(n) = 4*a(n-2)-6*a(n-4)+4*a(n-6)-a(n-8), for n>7.

a(2*n) = A002411(n).

a(2*n+1) = A000330(n).

a(n) = n*(n*(5*n+6)+(n^2+6*n+2)*(-1)^n-2)/96. - Bruno Berselli, Apr 14 2011

Example

Suppose n=7, then S^<7>=[(1,1.8019..,2.2469..);(1,0.445..,-0.8019..);(1,-1.2469..,0.5549..)] and (S^<7>)^T*S^<7>=[(3,1,2);(1,5,3);(2,3,6)]. Hence a(7)=Trace((S^<7>)^T*S^<7>)=3+5+6=14. (Note that (S^<n>)^T*S^<n> is always integral.)

Prog

(Maxima) makelist(coeff(taylor(x^2*(x^3+2*x^2+x+1)/((x-1)*(x+1))^4, x, 0, n), x, n), n, 0, 50); /* Bruno Berselli, May 30 2011 */

Crossrefs

Sequence in context: A300750 A101493 A347276 * A039668 A281175 A281221

Adjacent sequences: A188064 A188065 A188066 * A188068 A188069 A188070

Keyword

nonn,easy

Author

L. Edson Jeffery, Apr 11 2011

Status

approved