A343233 Triangle read by rows: Riordan triangle T = (1 - x*c(x), x), with the generating function c of A000108 (Catalan).

1, -1, 1, -1, -1, 1, -2, -1, -1, 1, -5, -2, -1, -1, 1, -14, -5, -2, -1, -1, 1, -42, -14, -5, -2, -1, -1, 1, -132, -42, -14, -5, -2, -1, -1, 1, -429, -132, -42, -14, -5, -2, -1, -1, 1, -1430, -429, -132, -42, -14, -5, -2, -1, -1, 1

Offset: 0

Author

Gary W. Adamson and Wolfdieter Lang, Apr 12 2021

Keywords

• sign
• tabl
• easy

Comments

As an unsigned sequence a(n) this is identical with the one of A155586(n+1), for n >= 0, but the triangle is not a simple signed version of A155586. See the formula.

This lower triangular Riordan matrix T of Toeplitz type is the inverse of the Riordan matrix (c(x), x) = |A106270|, also of Toeplitz type.

Examples

			The triangle matrix T begins:
  n/m     0    1    2   3   4   5   6   7   8   9 ...
  --------------------------------------------------
  0:      1
  1:     -1    1
  2:     -1   -1    1
  3:     -2   -1   -1   1
  4:     -5   -2   -1  -1   1
  5:    -14   -5   -2  -1  -1   1
  6:    -42  -14   -5  -2  -1  -1   1
  7:   -132  -42  -14  -5  -2  -1  -1   1
  8:   -429 -132  -42 -14  -5  -2  -1  -1   1
  9:  -1430 -429 -132 -42 -14  -5  -2  -1  -1   1
  ...
		

Crossrefs

Cf. A106270 (unsigned), A155586.

Formula

The lower triangular matrix T satisfies: T = I - L^{tr}*|A106270|, also for the finite N X N version, with the unit matrix I and the lower triangular matrix L^{tr}(i, j) = delta_{i, j-1} (Kronecker symbol delta) with first lower diagonal of 1s and 0 otherwise.

T(n, n) = 1, and for T(n, m) = -C_{n - 1 - m } = - |A106270(n-1, m)|, for 0 <= m <= n-1, with the Catalan numbers C(n) = A000108, and T(n, m) = 0 for n < m.

O.g.f. of column m: (1/c(x))*x^m = (1 - x*c(x))*x^m (Riordan matrix of Toeplitz type), with the o.g.f. c of A000108.

O.g.f. row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m, that is the o.g.f. of the triangle. G(z, x) = c(z)/(1 - x*z).