Search: id:A146061
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%I A146061
%S A146061 1,0,1,1,0,1,1,1,0,2,1,1,1,0,2,1,1,1,2,0,3,1,1,1,2,2,0,4,2,1,1,
%T A146061 2,2,3,0,5,2,2,1,2,2,3,4,0,6,2,2,2,2,2,3,4,5,0,8,2,2,2,4,2,3,
%U A146061 4,5,6,0,10,3,2,2,4,4,3,4,5,6,8,0,12,3,3,2,4,4,6,4,5,6,8,10
%V A146061 1,0,1,1,0,1,-1,1,0,2,1,-1,1,0,2,-1,1,-1,2,0,3,1,-1,1,-2,2,0,4,-2,1,-1,
%W A146061 2,-2,3,0,5,2,-2,1,-2,2,-3,4,0,6,-2,2,-2,2,-2,3,-4,5,0,8,2,-2,2,-4,2,-3,
%X A146061 4,-5,6,0,10,-3,2,-2,4,-4,3,-4,5,-6,8,0,12,3,-3,2,-4,4,-6,4,-5,6,-8,10
%N A146061 Eigentriangle, row sums = A000009, the number of partitions of n into 
               odd parts.
%C A146061 Right border = A000009; row sums = A000009 with offset 1.
%C A146061 Sum of n-th row terms = rightmost term in next row.
%C A146061 The INVERTi transform of A000009 starting with offset 1 = (1, 0, 1, -1, 
               -1,
%C A146061 1, -2, 2, -2, 2, -3, 3, -3, 4, -5, 5, -5, 6,...); i.e. A000700 signed 
               = left border.
%C A146061 A000700 is derived from parity changes of A000041 as follows: Given A000041: 
               (1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135,...). Write 
               down the parity starting (1, 1, 0, 1, 1, 1, 1, 1...) then add "1" 
               starting in the next
%C A146061 string of A000041 with a change in parity. Since the next 4 terms of 
               A000041
%C A146061 are (22, 30, 42, 56...) we denote these by (...2, 2, 2, 2...). The next 
               three p(n)
%C A146061 terms are 77, 101, 135, so these are (...3, 3, 3,...) in A000700.
%C A146061 The signed version of A000700 as indicated: (alternate signs starting 
               with
%C A146061 A000700(3): (+-+...) = the INVERTi transform of A000009.
%F A146061 Let M = triangle by columns: A000700 (signed, starting 1, 0, 1, -1, 1, 
               -1, 1, -2,...)
%F A146061 in every column and P = an infinite lower triangular matrix with
%F A146061 A000009 (1, 1, 1, 2, 2, 3, 4, 5, 6,...) as the right border and the rest 
               zeros.
%F A146061 A146061 = M * P
%e A146061 First few rows of the triangle =
%e A146061 1;
%e A146061 0, 1;
%e A146061 1, 0, 1;
%e A146061 -1, 1, 0, 2;
%e A146061 1, -1, 1, 0, 2;
%e A146061 -1, 1, -1, 2, 0, 3;
%e A146061 1, -1, 1, -2, 2, 0, 4;
%e A146061 -2, 1, -1, 2, -2, 3, 0, 5;
%e A146061 2, -2, 1, -2, 2, -3, 4, 0, 6;
%e A146061 -2, 2, -2, 2, -2, 3, -4, 5, 0, 8;
%e A146061 2, -2, 2, -4, 2, -3, 4, -5, 6, 0, 10;
%e A146061 -3, 2, -2, 4, -4, 3, -4, 5, -6, 8, 0, 12;
%e A146061 3, -3, 2, -4, 4, -6, 4, -5, 6, -8, 10, 0, 15;
%e A146061 -3, 3, -3, 4, -4, 6, -8, 5, -6, 8, -10, 12, 0, 18;
%e A146061 4, -3, 3, -6, 4, -6, 8, -10, 6, -8, 10, -12, 15, 0, 22;
%e A146061 -5, 4, -3, 6, -6, 6, -8, 10, -12, 8, -10, 12, -15, 18, 0, 27;
%e A146061 5, -5, 4, -6, 6, -9, 8, -10, 12, -16, 10, -12, 15, -18, 22, 0, 32;
%e A146061 ...
%Y A146061 A000009, Cf. A000700
%Y A146061 Sequence in context: A026609 A090340 A117162 this_sequence A135936 A109707 
               A064272
%Y A146061 Adjacent sequences: A146058 A146059 A146060 this_sequence A146062 A146063 
               A146064
%K A146061 tabl,sign
%O A146061 1,10
%A A146061 Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2008

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