A091698 Matrix inverse of triangle A063967.

1, -1, 1, 1, -3, 1, -1, 8, -5, 1, 1, -23, 19, -7, 1, -1, 74, -69, 34, -9, 1, 1, -262, 256, -147, 53, -11, 1, -1, 993, -986, 615, -265, 76, -13, 1, 1, -3943, 3935, -2571, 1235, -431, 103, -15, 1, -1, 16178, -16169, 10862, -5591, 2216, -653, 134, -17, 1, 1

Offset: 0

Author

Christian G. Bower, Jan 29 2004

Keywords

• sign
• tabl

Comments

Riordan array (1/(1+x), (sqrt(1+6x+5x^2)-x-1)/(2(1+x))). The absolute value array is (1/(1-x),xc(x)/(1-xc(x))) where c(x) is the g.f. of A000108. It factorizes as (1/(1-x),x/(1-x))(1,xc(x)). - Paul Barry, Jun 10 2005

Examples

			From _Paul Barry_, Apr 15 2010: (Start)
Triangle begins
  1,
  -1, 1,
  1, -3, 1,
  -1, 8, -5, 1,
  1, -23, 19, -7, 1,
  -1, 74, -69, 34, -9, 1,
  1, -262, 256, -147, 53, -11, 1,
  -1, 993, -986, 615, -265, 76, -13, 1,
  1, -3943, 3935, -2571, 1235, -431, 103, -15, 1
Production matrix begins
  -1, 1,
  0, -2, 1,
  0, 1, -2, 1,
  0, -1, 1, -2, 1,
  0, 1, -1, 1, -2, 1,
  0, -1, 1, -1, 1, -2, 1,
  0, 1, -1, 1, -1, 1, -2, 1,
  0, -1, 1, -1, 1, -1, 1, -2, 1,
  0, 1, -1, 1, -1, 1, -1, 1, -2, 1,
  0, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1 (End)
		

Links

• Lara K. Pudwell, Ascent sequences and the binomial convolution of Catalan numbers, arXiv preprint arXiv:1408.6823 [math.CO], 2014.

Crossrefs

Row sums: A091699. Row sums (absolute values): A007317. Column 1: A050511.

Programs

Mathematica

rows = 11; t[n_, k_] := Sum[Binomial[j, n - j]*Binomial[j, k], {j, 0, n}]; T = Table[t[n, k], {n, 0, rows - 1}, {k, 0, rows - 1}] // Inverse; Table[ T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 11 2017 *)