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%I #2 Mar 30 2012 17:25:32
%S 1,1,-1,-1,2,0,-1,3,0,-1,1,-4,0,4,1,1,-5,0,10,5,-2,-1,6,0,-20,-15,12,
%T 9,-1,7,0,-35,-35,42,63,9,1,-8,0,56,70,-112,-252,-72,50,1,-9,0,84,126,
%U -252,-756,-324,450,267,-1,10,0,-120,-210,504,1890,1080,-2250,-2670
%N Triangle, row sums = a signed, shifted version of A000587, the Rao Uppuluri-Carpenter numbers.
%C Right border = A000587, the Rao Uppuluri-Carpenter numbers, with different signs: (1, 1, 0, 1, 1, 2, 9, -9, 50, -267, -413, -2180,...).
%C Row sums = the same sequence shifted: (1, 0, 1, 1, 2, 9,...).
%F Let A = the self-inverse triangle, A118433. Shift the triangle down one place placing "1" at (0,0). Lim(n..->inf.) A^n, = a signed version B of the Rao Uppuluri-Carpenter numbers (A000587), as follows: (1, 1, 0, 1, 1, 2, 9, -9, 50, -267, -413, -2180,...). A000587 = (1, -1, 0, 1, 1, -2, -9, -9, 50, 267, 413, -2180,...). Triangle A144185 = (A * (an infinite lower triangular matrix with B as the main diagonal and the rest zeros)). These operations are equivalent to (by rows), taking termwise products of A118433 row terms and B, the signed version of the Rao Uppuluri-Carpenter numbers.
%e First few rows of the triangle =
%e 1;
%e 1, -1;
%e -1, 2, 0;
%e -1, 3, 0, -1,
%e 1, -4, 0, 4, 1;
%e 1, -5, 0, 10, 5, -2;
%e -1, 6, 0, -20, -15, 12, 9;
%e -1, 7, 0, -35, -35, 42, 63, 9;
%e 1, -8, 0, 56, 70, -112, -252, -72, 50;
%e 1, -9, 0, 84, 126, -252, -756, -324, 450, 267;
%e -1, 10, 0, -120, -210, 504, 1890, 1080, -2250, -2670, -413;
%e ...
%e Example: row 5 = (1, -5, 0, 10, 5, -2) = termwise products of row 5 of the self-inverse triangle A118433: (1, -5, -10, 10, 5, -1) and the first 6 terms of the "B" signed version of A000587 (the rao Uppuluri-Carpenter numbers): (1, 1, 0, 1, 1, 2) = (1*1, -5*1, 0*0, 10*1, 5*1, -5*2).
%Y A000587, Cf. A118434
%K tabl,sign
%O 0,5
%A _Gary W. Adamson_, Sep 13 2008
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