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%I #10 Mar 13 2015 00:49:42
%S 1,1,1,0,1,2,-1,0,2,3,-1,-1,0,3,4,0,-1,-2,0,4,5,1,0,-2,-3,0,5,6,1,1,0,
%T -3,-4,0,6,7,0,1,2,0,-4,-5,0,7,8,-1,0,2,3,0,-5,-6,0,8,9,-1,-1,0,3,4,0,
%U -6,-7,0,9,10,0,-1,-2,0,4,5,0,-7,-8,0,10,11,1,0,-2,-3,0,5,6,0,-8,-9,0
%N Eigentriangle generated from inverse of 6th cyclotomic polynomial, row sums = n+1.
%C Left border = A010892. Right border = (1, 1, 2, 3, 4,...), row sums = (1, 2, 3,...).
%F Eigentriangle by rows, T(n,k) = A010892(n-k)*A000027(k-1). A010892 = the inverse of the 6th cyclotomic polynomial: (1, 1, 0, -1, -1, 0,...); and A000027(k-1) = (1, 2, 3,...) offset, = (1, 1, 2, 3,...).
%F Let A = an infinite lower triangular decrescendo subsequences matrix with A010892: (1, 1, 0, -1, -1, 0,...) in every column: (1; 1,1; 0,1,1; -1,0,1,1;...); let B = an infinite lower triangular matrix with (1,1,2,3,...) in the main diagonal and the rest zeros; then A144082 = A*B.
%e First few rows of the triangle =
%e 1;
%e 1, 1;
%e 0, 1, 2;
%e -1, 0, 2, 3;
%e -1, -1, 0, 3, 4;
%e 0, -1, -2, 0, 4, 5;
%e 1, 0, -2, -3, 0, 5, 6;
%e 1, 1, 0, -3, -4, 0, 6, 7;
%e 0, 1, 2, 0, -4, -5, 0, 7, 8;
%e -1, 0, 2, 3, 0, -5, -6, 0, 8, 9;
%e -1, -1, 0, 3, 4, -6, -7, 0, 9, 10;
%e 0, -1, -2, 0, 4, 5, 0, -7, -8, 0, 10, 11;
%e 1, 0, -2, -3, 0, 5, 6, 0, -8, -9, 0, 11, 12;
%e ...
%e Example: row 3 = (-1, 0, 3, 4) = termwise product of (-1, 0, 1, 1) and (1, 2, 3, 4) = (-1*1, 0*2, 1*3, 1*4). (-1, 0, 1, 1) = first 4 terms of A010892 reversed.
%Y Cf. A000027, A010892.
%K tabl,sign
%O 0,6
%A _Gary W. Adamson_, Sep 10 2008
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