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

1,10

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.

Links

Table of n, a(n) for n=1..89.

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.

Example

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

Sequence in context: A340676 A117162 A277045 * A331186 A372504 A135936

Adjacent sequences: A146058 A146059 A146060 * A146062 A146063 A146064

Keyword

tabl,sign

Author

Gary W. Adamson, Oct 26 2008

Status

approved