cp's OEIS Frontend

A146061 Eigentriangle, row sums = A000009, the number of partitions of n into odd parts.

%I A146061 #6 Apr 18 2013 10:04:26
%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,
%T A146061 -1,2,-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,
%U A146061 2,-3,4,-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, 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 string of A000041 with a change in parity. Since the next 4 terms of A000041 are (22, 30, 42, 56...) we denote these by (...2, 2, 2, 2...). The next three p(n) 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 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,...) in every column and P = an infinite lower triangular matrix with A000009 (1, 1, 1, 2, 2, 3, 4, 5, 6,...) as the right border and the rest zeros. 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
%K A146061 tabl,sign
%O A146061 1,10
%A A146061 _Gary W. Adamson_, Oct 26 2008