|
%I #7 Nov 16 2012 12:27:54
%S -8,6,-27,65,-56,15,-61,260,-469,415,-176,28,-114,736,-2104,3214,
%T -2838,1456,-400,45,-190,1714,-6988,15699,-21461,18760,-10614,3768,
%U -760,66,-293,3507,-19195,58807,-112123,141441,-122168,73185,-30077,8107,-1288,91
%N 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.
%C The row length of this irregular triangle is 2*(m-1), m >= 2.
%C 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)).
%C 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)).
%C Using inclusion-exclusion one can write (x^2-4)^m*P(m;2,x^2) =
%C 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.
%F 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.
%e The array a(m,k) starts:
%e m\k 0 1 2 3 4 5 6 7 8 9 ...
%e 2: -8 6
%e 3: -27 65 -56 15
%e 4: -61 260 -469 415 -176 28
%e 5: -114 736 -2104 3214 -2838 1456 -400 45
%e 6: -190 1714 -6988 15699 -21461 18760 -10614 3768 -760 66
%e ...
%e Row m=7: -293, 3507, -19195, 58807, -112123, 141441, -122168, 73185, -30077, 8107, -1288, 91.
%e Row m=8: -427, 6536, -46102, 183762, -461654, 780716, -926345, 790773, -491397, 221760, -71139, 15405, -2016, 120.
%e Row 9: -596, 11346, -100077, 502036, -1600280, 3470116, -5352805, 6051236, -5110145, 3256825, -1568416, 564980, -148176, 26770, -2976, 153.
%e 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).
%e 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.
%Y Cf. A217478.
%K sign,easy,tabf
%O 2,1
%A _Wolfdieter Lang_, Nov 14 2012
|
