A251635 Riordan array (1-2*x,x), inverse of Riordan array (1/(1-2*x), x) = A130321.

1, -2, 1, 0, -2, 1, 0, 0, -2, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0

Offset: 0

Author

Wolfdieter Lang, Jan 10 2015

Keywords

• sign
• easy
• tabl

Comments

This is a simple Riordan array, an infinite lower triangular matrix. It is the inverse matrix of A130321 (with zeros above the diagonal).

Row sums have o.g.f. (1-2*x)/(1-x) and give 1, repeat(-1), i.e., A153881(n+1), n >= 0.

Alternate row sums have o.g.f. (1-2*x)/(1+x) and give 1, repeat(-3,3), i.e., (-1)^n*A122553(n).

Examples

			The triangle T(n, k) begins:
n\k  0  1  2  3  4  5  6  7  8  9 10 ...
0:   1
1:  -2  1
2:   0 -2  1
3:   0  0 -2  1
4:   0  0  0 -2  1
5:   0  0  0  0 -2  1
6:   0  0  0  0  0 -2 1
7:   0  0  0  0  0  0 -2  1
8:   0  0  0  0  0  0  0 -2  1
9:   0  0  0  0  0  0  0  0 -2  1
10:  0  0  0  0  0  0  0  0  0 -2  1
...
		

Crossrefs

Cf. A130321, A153881, A122553.

Programs

Haskell

a251635 n k = a251635_tabl !! n !! k
a251635_row n = a251635_tabl !! n
a251635_tabl = [1] : iterate (0 :) [-2, 1]
-- Reinhard Zumkeller, Jan 11 2015

Formula

T(n, k) = 0 if n < k and k = 0..(n-2) for n >= 2, and T(n, n) = 1 and T(n, n-1) = -2.

G.f. for row polynomials P(n, x) = -2^x^(n-1) + x^n is (1-2*z)/(1-x*z).

G.f. for k-th column: (1-2*x)*x^k, k >= 0.