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A136203 Derived Shabat linear tree transform of A053120: Triangle of coefficients of transformed Chebyshev's T(n, x) polynomials (powers of x in increasing order) T(x,n)->c*T(c*x+d)+d: c=-1;d=1; as substitution: 1-x->y( here alternative starting polynomial of Q(y,1]=1-y. 0

%I

%S 1,1,-1,1,-2,2,1,-3,8,-4,1,-4,20,-24,8,1,-5,40,-84,64,-16,1,-6,70,

%T -224,288,-160,32,1,-7,112,-504,960,-880,384,-64,1,-8,168,-1008,2640,

%U -3520,2496,-896,128,1,-9,240,-1848,6336,-11440,11648,-6720,2048,-256,1,-10,330,-3168,13728,-32032,43680,-35840,17408

%N Derived Shabat linear tree transform of A053120: Triangle of coefficients of transformed Chebyshev's T(n, x) polynomials (powers of x in increasing order) T(x,n)->c*T(c*x+d)+d: c=-1;d=1; as substitution: 1-x->y( here alternative starting polynomial of Q(y,1]=1-y.

%C Row sums are:

%C {1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1}

%C integration is alternating on transformed {0,2} domain:

%C Table[Table[Integrate[Sqrt[1/(1 - (1 - y)^2)]*Q[y, n]*Q[y, m], {y, 0, 2}], {n, 0, 10}], {m, 0, 10}]

%D http://logic.pdmi.ras.ru/~yumat/personaljournal/chebyshev/chebysh.htm Quote: "... how many polynomials can generate given tree? It is easy to see that if P is a generalized Chebyshev Polynomial, then so is polynomial CP(cz+d)+D, moreover, it represents the same tree (of course, provided that both C and c are different from zero). In some natural sense these two linear transformations exhaust the variety of polynomials representing given tree. Namely, every drawing of a tree on the plane introduces an additional structure--circular order of edges around given vertex (say, clock-wise). Dealing with Chebyshev polynomials, it is natural to speak about plane trees understanding by them trees with this additional structure. "

%F Q(y,0)=1;Q(y,1)=1-y; Q(y, n) = -(-2 + 2 (1 - y) - 2 (1 - y) Q(y, n - 1) + Q(y, n - 2))

%e {1},

%e {1, -1},

%e {1, -2, 2},

%e {1, -3, 8, -4},

%e {1, -4, 20, -24,8},

%e {1, -5, 40, -84, 64, -16},

%e {1, -6, 70, -224, 288, -160, 32},

%e {1, -7, 112, -504, 960, -880, 384, -64},

%e {1, -8, 168, -1008, 2640, -3520, 2496, -896, 128},

%e {1, -9, 240, -1848, 6336, -11440, 11648, -6720, 2048, -256},

%e {1, -10, 330, -3168, 13728, -32032, 43680, -35840, 17408, -4608, 512}

%t Clear[c, d, x0, x1, x2, P, Q, x, n, a] P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; Solve[c*x0 + d - 2*x*(c*x1 + d) + c*x2 + d == 0, x0] c = -1; d = 1; (* Transform : c*x + d -> y*) Q[y, -1] = 0; Q[y, 0] = 1; Q[y, 1] =1- y; Q[y_, n_] := Q[y, n] = -(-2 + 2 (1 - y) - 2 (1 - y) Q[y, n - 1] + Q[y, n - 2]); Table[ExpandAll[Q[y, n]], {n, 0, 10}]; a = Table[CoefficientList[Q[y, n], y], {n, 0, 10}]; Flatten[a]

%Y Cf. A053120.

%K uned,tabl,sign

%O 1,5

%A _Roger L. Bagula_, Mar 16 2008

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