A144185 Triangle, row sums = a signed, shifted version of A000587, the Rao Uppuluri-Carpenter numbers.

1, 1, -1, -1, 2, 0, -1, 3, 0, -1, 1, -4, 0, 4, 1, 1, -5, 0, 10, 5, -2, -1, 6, 0, -20, -15, 12, 9, -1, 7, 0, -35, -35, 42, 63, 9, 1, -8, 0, 56, 70, -112, -252, -72, 50, 1, -9, 0, 84, 126, -252, -756, -324, 450, 267, -1, 10, 0, -120, -210, 504, 1890, 1080, -2250, -2670

Offset: 0

Author

Gary W. Adamson, Sep 13 2008

Keywords

• tabl
• sign

Comments

Right border = A000587, the Rao Uppuluri-Carpenter numbers, with different signs: (1, 1, 0, 1, 1, 2, 9, -9, 50, -267, -413, -2180,...).

Row sums = the same sequence shifted: (1, 0, 1, 1, 2, 9,...).

Let A = the self-inverse triangle, A118433. Shift the triangle down one place placing "1" at (0,0). Lim_{n->oo} A^n, = a signed version B of the Rao Uppuluri-Carpenter numbers (A000587), as follows: (1, 1, 0, 1, 1, 2, 9, -9, 50, -267, -413, -2180,...). This triangle = (A * (an infinite lower triangular matrix with B as the main diagonal and the rest zeros)). These operations are equivalent to (by rows), taking termwise products of A118433 row terms and B, the signed version of the Rao Uppuluri-Carpenter numbers.

Examples

			First few rows of the triangle =
  1;
  1, -1;
  -1, 2, 0;
  -1, 3, 0, -1,
  1, -4, 0, 4, 1;
  1, -5, 0, 10, 5, -2;
  -1, 6, 0, -20, -15, 12, 9;
  -1, 7, 0, -35, -35, 42, 63, 9;
  1, -8, 0, 56, 70, -112, -252, -72, 50;
  1, -9, 0, 84, 126, -252, -756, -324, 450, 267;
  -1, 10, 0, -120, -210, 504, 1890, 1080, -2250, -2670, -413;
...
Example: row 5 = (1, -5, 0, 10, 5, -2) = termwise products of row 5 of the self-inverse triangle A118433: (1, -5, -10, 10, 5, -1) and the first 6 terms of the "B" signed version of A000587 (the Rao Uppuluri-Carpenter numbers): (1, 1, 0, 1, 1, 2) = (1*1, -5*1, 0*0, 10*1, 5*1, -5*2).
		

Crossrefs

Cf. A000587, A118433, A118434.