A174739 Triangle read by rows, a partition number generator; A145006 * the diagonalized variant of A000041, (A174712).

1, 1, 1, 0, 1, 2, 0, 0, 2, 3, -1, 0, 0, 3, 5, 0, -1, 0, 0, 5, 7, -1, 0, -2, 0, 0, 7, 11, 0, -1, 0, -3, 0, 0, 11, 15, 0, 0, -2, 0, -5, 0, 15, 22, 0, 0, 0, -3, 0, -7, 0, 0, 22, 30, 0, 0, 0, 0, -5, 0, -11, 0, 0, 30, 42, 1, 0, 0, 0, 0, -7, 0, -15, 0, 0, 42, 56, 0, 1, 0, 0, 0, 0, -11, 0, -22, 0, 0, 0, 0

Offset: 1

Author

Gary W. Adamson, Mar 28 2010

Keywords

• tabl
• sign

Comments

Row sums = the partition numbers, A000041 starting with offset 1.

The triangle demonstrates an equivalency to Euler's pentagonal recurrence

relation, such that sum of n-th row terms = rightmost term of next row, a

partition number.

Contribution from Gary W. Adamson, Mar 28 2010: (Start)

A174739 is equivalent to Euler's pentagonal theorem in triangular form.

For example, row 9 = (0, 0, -2, 0, -5, 0, 0, 15, 22) or: p(9) = 30 = p(8)

+ p(7) - p(4) - p(2). (End)

Examples

			First few rows of the triangle =
1;
1, 1;
0, 1, 2;
0, 0, 2, 3;
-1, 0, 0, 3, 5;
0, -1, 0, 0, 5, 7;
-1, 0, -2, 0, 0, 7, 11;
0, -1, 0, -3, 0, 0, 11, 15;
0, 0, -2, 0, -5, 0, 0, 15, 22;
0, 0, 0, -3, 0, -7, 0, 0, 22, 30;
0, 0, 0, 0, -5, 0, -11, 0, 0, 30, 42;
1, 0, 0, 0, 0, -7, 0, -15, 0, 0, 42, 56;
0, 1, 0, 0, 0, 0, -11, 0, 22, 0, 0, 56, 77;
0, 0, 2, 0, 0, 0, 0, -15, 0, -30, 0, 0, 77, 101;
1, 0, 0, 3, 0, 0, 0, 0, -22, 0, -42, 0, 0, 101, 135;
0, 1, 0, 0, 5, 0, 0, 0, 0, -30, 0, -56, 0, 0, 135, 176;
0, 0, 2, 0, 0, 7, 0, 0, 0, 0, -42, 0, -77, 0, 0, 176, 231;
0, 0, 0, 3, 0, 0, 11, 0, 0, 0, 0, -56, 0, -101, 0, 0, 231, 297;
...
		

Crossrefs

Cf. A000041, A145006, A174712

Formula

Given triangle A145006, delete the first "1", = triangle Q. With M = A174712,

the diagonalize variant of the partition numbers, perform Q*M as infinite lower

triangular matrices.