A127672: Table of coefficients of Chebyshev T-polynomials with scaled argument. Increasing powers of y, with zeros. a(n,m) tabl head (triangle) for A127672. Scaled coefficient triangle for Chebyshev's T(n,x) (increasing scaled powers). T(n,x)= sum(a(n,m)*(2^(m-1))*x^m,m=0..n). n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 -3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 2 0 -4 0 1 0 0 0 0 0 0 0 0 0 0 0 5 0 5 0 -5 0 1 0 0 0 0 0 0 0 0 0 0 6 -2 0 9 0 -6 0 1 0 0 0 0 0 0 0 0 0 7 0 -7 0 14 0 -7 0 1 0 0 0 0 0 0 0 0 8 2 0 -16 0 20 0 -8 0 1 0 0 0 0 0 0 0 9 0 9 0 -30 0 27 0 -9 0 1 0 0 0 0 0 0 10 -2 0 25 0 -50 0 35 0 -10 0 1 0 0 0 0 0 11 0 -11 0 55 0 -77 0 44 0 -11 0 1 0 0 0 0 12 2 0 -36 0 105 0 -112 0 54 0 -12 0 1 0 0 0 13 0 13 0 -91 0 182 0 -156 0 65 0 -13 0 1 0 0 14 -2 0 49 0 -196 0 294 0 -210 0 77 0 -14 0 1 0 15 0 -15 0 140 0 -378 0 450 0 -275 0 90 0 -15 0 1 . . . Row sums (signed): [2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, -1, -2,...]= A057079(n-1). Row sums (unsigned): [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521,...]= A000032(n) (Lucas numbers). Bisection: Triangle of even numbered rows (without zeros): A127677. Unsigned triangle of odd numbered rows (without zeros): A111125. The polynomials, written in the scaled variable x are, for n=0..15: n=0: 2 n=1: x n=2: -2+x^2 n=3: -3*x+x^3 n=4: 2-4*x^2+x^4 n=5: 5*x-5*x^3+x^5 n=6: -2+9*x^2-6*x^4+x^6 n=7: -7*x+14*x^3-7*x^5+x^7 n=8: 2-16*x^2+20*x^4-8*x^6+x^8 n=9: 9*x-30*x^3+27*x^5-9*x^7+x^9 n=10: -2+25*x^2-50*x^4+35*x^6-10*x^8+x^10 n=11: -11*x+55*x^3-77*x^5+44*x^7-11*x^9+x^11 n=12: 2-36*x^2+105*x^4-112*x^6+54*x^8-12*x^10+x^12 n=13: 13*x-91*x^3+182*x^5-156*x^7+65*x^9-13*x^11+x^13 n=14: -2+49*x^2-196*x^4+294*x^6-210*x^8+77*x^10-14*x^12+x^14 n=15: -15*x+140*x^3-378*x^5+450*x^7-275*x^9+90*x^11-15*x^13+x^15 etc. ########################## e.o.f. ###################################