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

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, 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, 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

Offset: 1

Author

Gary W. Adamson, Oct 26 2008

Keywords

• tabl
• sign

Comments

Right border = A000009; row sums = A000009 with offset 1.

Sum of n-th row terms = rightmost term in next row.

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.

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.

The signed version of A000700 as indicated: (alternate signs starting with A000700(3): (+-+...) = the INVERTi transform of A000009.

Examples

			First few rows of the triangle =
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, 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, 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, 0, 15;
-3, 3, -3, 4, -4, 6, -8, 5, -6, 8, -10, 12, 0, 18;
4, -3, 3, -6, 4, -6, 8, -10, 6, -8, 10, -12, 15, 0, 22;
-5, 4, -3, 6, -6, 6, -8, 10, -12, 8, -10, 12, -15, 18, 0, 27;
5, -5, 4, -6, 6, -9, 8, -10, 12, -16, 10, -12, 15, -18, 22, 0, 32;
...
		

Crossrefs

A000009, Cf. A000700

Formula

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.