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%I #3 Apr 09 2011 18:21:28
%S 1,1,1,1,-1,1,1,1,2,0,-2,1,-1,1,1,1,1,1,3,1,-1,-3,3,-1,-1,3,1,-1,1,-1,
%T 1,1,1,1,1,4,2,0,-2,-4,6,0,-2,0,6,4,-2,0,2,-4,1,-1,1,-1,1,1,1,1,1,1,1,
%U 5,3,1,-1,-3,-5,10,2,-2,-2,2,10,10,-2,-2,2,2,-10,5,-3,1,1,-3,5,1,-1,1,-1,1,-1
%N Matrices C_n (read by rows), defined by the following identity: C_n * P(X,Y) = P(X+Y, X-Y), where P is any homogeneous polynomial of degree n in two variables, represented as a column whose i-th element is the coefficient of X^(n-i)Y^i.
%F Formula: (C_n)ij = Sum(k=0..j, (-1)^k * C(j, k) * C(n-j, i-k)).
%F Recurrence: (C_n)i,j = (C_(n-1))i,j + (C_(n-1))i-1,j for all j < n; (C_n)i,j = (C_(n-1))i,j-1 - (C_(n-1))i-1,j-1 for all j > 0.
%e C_0 = [1]
%e C_1 = [1 1 / 1 -1]
%e C_2 is equal to:
%e |1 1 1|
%e |2 0 -2|
%e |1 -1 1|
%e since if
%e P(X,Y) = a0*X^2 + a1*XY + a2*Y^2
%e then
%e P(X+Y,X-Y) = (a0 + a1 + a2)*X^2 + (2*a0 - 2*a2)*XY + (a0 - a1 + a2)*Y^2
%K easy,tabf,sign
%O 1,9
%A Ilia Smilga (ilia.smilga(AT)orange.fr), Feb 14 2008, Feb 20 2008
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