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%I #11 Mar 06 2018 18:50:55
%S 1,9,-6,1,25,-50,35,-10,1,49,-196,294,-210,77,-14,1,81,-540,1386,
%T -1782,1287,-546,135,-18,1,121,-1210,4719,-9438,11011,-8008,3740,
%U -1122,209,-22,1,169,-2366,13013,-37180,63206,-68952,50388,-25194,8645,-2002,299,-26,1
%N Coefficient array for powers of x^2 of the square of Chebyshev's C(2*n+1,x)/x =: tau(n,x) polynomials.
%C The row lengths sequence of this irregular triangle is 2*n+1 = A005408(n).
%C For the coefficient array of powers of x^2 of the monic integer Chebyshev C(2*n+1,x)/x = :tau(n,x) polynomials see the signed triangle ((-1)^(n-m))*A111125(n,m). See the comment from Oct 23 2012.
%C The o.g.f. of the row polynomials p(n,x) := sum(a(n,m)*x^m, m=0..2*n), n>=0, is G2(x,z) = sum(p(n,x)*z^n,n=0..infinity) = (1+ (6-2*x)*z+z^2)/((1-z)*((z+1)^2-z*(x-2)^2)). Derived from the odd part of the bisection of the o.g.f. for the C(n,x)^2 polynomials. Note that p(n,x) = (tau(n,sqrt(x)))^2.
%F a(n,m) = [x^m] (p(n,x)), n>=0, 0 <= m <= 2*n, with p(n,x) = (C(2*n+1,sqrt(x))/sqrt(x))^2 = (tau(n,sqrt(x)))^2, For Chebyshev's C and tau polynomials see a comment above.
%F For n >= 0, 0 <= m <= 2*n, a(n,m) = [x^m*z^n] G2(x,z), where the o.g.f. G2(x,z) given in a comment above.
%F a(n,m) = (-1)^m * A156308(2*n+1,m+1). - _Max Alekseyev_, Mar 06 2018
%e The array begins:
%e n\m 0 1 2 3 4 5 6 7 8 9 10
%e 0: 1
%e 1: 9 -6 1
%e 2: 25 -50 35 -10 1
%e 3: 49 -196 294 -210 77 -14 1
%e 4: 81 -540 1386 -1782 1287 -546 135 -18 1
%e 5: 121 -1210 4719 -9438 11011 -8008 3740 -1122 209 -22 1
%e ...
%e Row polynomial for n=1: p(1,x) = (tau(1,sqrt(x)))^2 = (3-1*x)^2 = 9 - 6*x +1*x^2.
%e Row polynomial for n=2: p(2,x) = (tau(2,sqrt(x)))^2 = (5 - 5*x + 1*x^2)^2 = 25 - 50*x + 35*x^2 - 10*x^3 + 1*x^4.
%Y Odd rows of A156308 with alternating signs of elements.
%Y Cf. A005408, A111125.
%K sign,easy,tabf
%O 0,2
%A _Wolfdieter Lang_, Jan 04 2013