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%I #7 Jan 28 2013 05:40:45
%S 1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,-1,0,0,1,1,0,0,-1,0,0,1,1,0,-1,0,-1,0,
%T 0,1,1,0,0,-1,0,-1,0,0,1,1,0,0,0,-1,0,-1,0,0,1,1,0,0,0,0,-1,0,-1,0,0,
%U 1,1,0,0,0,0,0,-1,0,-1,0,0,1,1,0
%N Triangle read by rows, generator for the partition numbers, A000041
%C The partition numbers, A000041, = eigenvector of the triangle. With A080995, characteristic function of the generalized pentagonal numbers, we apply signs: (++ -- ++,...) to the 1's, starting with offset 1. This gives an opposite parity to Euler's partition formula which is (with offset 1): -p(n-1) - p(n-2) + p(n-5) + p(n-7),...
%C By applying termwise products of A000041 terms and row terms of A145006, we obtain the eigentriangle of the partition numbers.
%F Triangle by columns: let A = an infinite lower triangular matrix with the characteristic function of A001318: (1, 2, 5, 7, 12, 15,...) in every column; signed: (++ -- ++,...).
%F Shift triangle A down one place and insert "1" in the T(0,0) position, giving triangle A145006. The eigenvector of the triangle = A000041, the partition numbers: (1, 1, 2, 3, 5, 7, 11,...). Lim_{n=1..inf} A145006^n = A000041. Or, simply take a suitably large power of the triangle, which quickly converges to A000041 as a vector.
%e First few rows of the triangle =
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 0, 1, 1, 0;
%e 0, 0, 1, 1, 0;
%e -1, 0, 0, 1, 1, 0;
%e 0, -1, 0, 0, 1, 1, 0;
%e -1, 0, -1, 0, 0, 1, 1, 0;
%e 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e 1, 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e ...
%Y A000041, Cf. A080995, A001318, A145007
%K eigen,tabl,sign
%O 0,1
%A _Gary W. Adamson_, Sep 28 2008
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