A145006 Triangle read by rows, generator for the partition numbers, A000041

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, 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, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0

Offset

0,1

Comments

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),...

By applying termwise products of A000041 terms and row terms of A145006, we obtain the eigentriangle of the partition numbers.

Links

Table of n, a(n) for n=0..77.

Formula

Triangle by columns: let A = an infinite lower triangular matrix with the characteristic function of A001318: (1, 2, 5, 7, 12, 15,...) in every column; signed: (++ -- ++,...).

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.

Example

First few rows of the triangle =
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, 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, 0;
0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
1, 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0;
...

Crossrefs

A000041, Cf. A080995, A001318, A145007

Sequence in context: A286801 A189091 A344863 * A189138 A213719 A286937

Adjacent sequences: A145003 A145004 A145005 * A145007 A145008 A145009

Keyword

eigen,tabl,sign

Author

Gary W. Adamson, Sep 28 2008

Status

approved