%I A145006
%S A145006 1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0,
%T A145006 1,1,0,0,1,0,1,0,0,1,1,0,0,0,1,0,1,0,0,1,1,0,0,0,0,1,0,1,0,0,1,1,
%U A145006 0,0,0,0,0,1,0,1,0,0,1,1,0
%V A145006 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,-1,0,0,
%W A145006 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,-1,0,0,1,1,
%X A145006 0,0,0,0,0,-1,0,-1,0,0,1,1,0
%N A145006 Triangle read by rows, generator for the partition numbers, A000041
%C A145006 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 A145006 By applying termwise products of A000041 terms and row terms of A145006, 
               we obtain the eigentriangle of the partition numbers.
%F A145006 Triangle by columns: let A = an an infinite lower triangular matrix with 
               the characteristic function of A001318: (1, 2, 5, 7, 12, 15,...) 
               in every column; signed: (++ -- ++,...).
%F A145006 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 A145006 First few rows of the triangle =
%e A145006 1;
%e A145006 1, 0;
%e A145006 1, 1, 0;
%e A145006 0, 1, 1, 0;
%e A145006 0, 0, 1, 1, 0;
%e A145006 -1, 0, 0, 1, 1, 0;
%e A145006 0, -1, 0, 0, 1, 1, 0;
%e A145006 -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 1, 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
%e A145006 ...
%Y A145006 A000041, Cf. A080995, A001318, A145007
%Y A145006 Sequence in context: A086747 A141727 A123594 this_sequence A080813 A100672 
               A079559
%Y A145006 Adjacent sequences: A145003 A145004 A145005 this_sequence A145007 A145008 
               A145009
%K A145006 eigen,tabl,sign
%O A145006 0,1
%A A145006 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 28 2008