A217479 Array of coefficients of polynomials providing the third term of the numerator of the generating function for odd powers (2*m+1) of Chebyshev S-polynomials. The present polynomials are called P(m;2,x^2), m >= 2.

-8, 6, -27, 65, -56, 15, -61, 260, -469, 415, -176, 28, -114, 736, -2104, 3214, -2838, 1456, -400, 45, -190, 1714, -6988, 15699, -21461, 18760, -10614, 3768, -760, 66, -293, 3507, -19195, 58807, -112123, 141441, -122168, 73185, -30077, 8107, -1288, 91

Offset: 2

Author

Wolfdieter Lang, Nov 14 2012

Keywords

• sign
• easy
• tabf

Comments

The row length of this irregular triangle is 2*(m-1), m >= 2.

For the o.g.f. of S(m,x)^(2*m+1), m>=0, with Chebyshev's S-polynomials (coefficient triangle A049310) see the comment on A217478. G(m;z,x) = Z(m;z,x)/N(m;z,x) with N(m;z,x) = product((1+z^2) - z*x*tau(k,x),k=0..m), and Z(m;z,x) = sum((1+z^2)^(m-l)*(-z*x)^l*P(m;l,x^2),l=0..m), where P(m,l,x^2) = sum(T(m,k)*S(2*k,x)*sigma(m;k,l,x^2), k=0..m)/(x^2-4)^m, with sigma(m;k,l,x^2) the elementary symmetric function of a product of l factors from tau(j,x), for j=0..m, with tau(k,x) missing. Here tau(j,x):= 2*T(2*j+1,x/2)/x = R(2*j+1,x)/x (see A127672 for the coefficients of R(n,x)).

The present array a(m,k) provides the P(m;2,x^2) coefficients, and m >= 2: P(m;2,x^2) = sum(a(m,k)*x^2,k=0..(2*m-3)).

Using inclusion-exclusion one can write (x^2-4)^m*P(m;2,x^2) =

sum(T(m,k)*S(2*k,x)*(sigma(m+1;2,x^2) - sum(tau(j,x),j=0..m)* tau(k,x) + tau(k,x)^2),k=0..m), with sigma(m+1;2,x^2) the elementary symmetric function of 2 factors from tau(j,x), for j=0,1,...,m. E.g.,m=2: sigma(2+1;2,x^2) = tau(0,x)*tau(1,x) + tau(0,x)*tau(2,x) + tau(1,x)*tau(2,x). The identities Id(0;m,x^2) and Id(1;m,x^2) (given in the comment on A217478) together with the new identity Id(2;m,x^2) := sum(T(m,k)*S(2*k,x)*tau(k,x)^2,k=0..m) = (x^2-4)^m*((x^2-1)^(2*m+1) + 1)/x^2 are now used. The new identity is obtained from the de Moivre-Binet formula for S and tau using first twice the identity mentioned in a Nov 14 2012 comment on A113187, and then the identity q^3 - 1/q^3 = sqrt(x^2-4)*(x^2-1) (see the instance k=1 of the formula given in a Oct 18 2012 comment on A111125 with x -> q which is defined by (x+sqrt(x^2-4))/2). This yields, after division by (x^2-4)^m, finally the polynomial P(2;m,x^2) = sigma(m+1;2,x^2) - sum(tau(j,x),j=0..m)*x^(2*m) + ((x^2-1)^(2*m+1) + 1)/x^2, for m >= 2.

Examples

			The array a(m,k) starts:
m\k   0    1     2     3      4     5      6    7    8  9 ...
2:   -8    6
3:  -27   65   -56    15
4:  -61  260  -469   415   -176    28
5: -114  736 -2104  3214  -2838  1456   -400   45
6: -190 1714 -6988 15699 -21461 18760 -10614 3768 -760 66
...
Row m=7:  -293, 3507, -19195, 58807, -112123, 141441, -122168, 73185, -30077, 8107, -1288, 91.
Row m=8: -427, 6536, -46102, 183762, -461654, 780716, -926345, 790773, -491397, 221760, -71139, 15405, -2016, 120.
Row 9: -596, 11346, -100077, 502036, -1600280, 3470116, -5352805, 6051236, -5110145, 3256825, -1568416, 564980, -148176, 26770, -2976, 153.
m=2: P(2;2,x^2) = tau(0,x)*tau(1,x) + tau(0,x)*tau(2,x) + tau(1,x)*tau(2,x) - (tau(0,x)+tau(1,x)+tau(2,x))*x^4 + (5 -10*x^2 + 10*x^4 - 5*x^6 + x^8)  = -8 + 6*x^2 = 2*(-4 + 3*x^2).
  The numerator of the o.g.f. for S(n,x)^5 is Z(2;z,x) = (1+z^2)^2 + (1+z^2)*(-x*z)*(3-4*x^2) + (-x*z)^2*2*(-4 + 3*x^2), where the last bracket in the second term comes from row m=2 of A217478. The denominator is N(2;z,x) = product((1+z^2)-z*x*tau(k,x), k=0..2). See the example of A217478.
		

Crossrefs

Cf. A217478.

Formula

a(m,k) = [x^(2*k)] P(2;m,x^2), m >= 2, k = 0..(2*m-3), with P(2;m,x^2) given in the comment above.