A113313 Riordan array (1-2x,x/(1-x)).

1, -2, 1, 0, -1, 1, 0, -1, 0, 1, 0, -1, -1, 1, 1, 0, -1, -2, 0, 2, 1, 0, -1, -3, -2, 2, 3, 1, 0, -1, -4, -5, 0, 5, 4, 1, 0, -1, -5, -9, -5, 5, 9, 5, 1, 0, -1, -6, -14, -14, 0, 14, 14, 6, 1, 0, -1, -7, -20, -28, -14, 14, 28, 20, 7, 1, 0, -1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, 0, -1, -9, -35

Offset: 0

Author

Paul Barry, Oct 25 2005

Keywords

• easy
• sign
• tabl

Comments

Row sums are (1,-1,0,0,0,...) = 2*C(0,n) - C(1,n).

Diagonal sums are -2*0^n - F(n-4) with g.f. (1 - 3x + 2x^2) / (1 - x - x^2).

Inverse of A113310.

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 -1  1
3:  0 -1  0   1
4:  0 -1 -1   1   1
5:  0 -1 -2   0   2   1
6:  0 -1 -3  -2   2   3  1
7:  0 -1 -4  -5   0   5  4  1
8:  0 -1 -5  -9  -5   5  9  5  1
9:  0 -1 -6 -14 -14   0 14 14  6 1
10: 0 -1 -7 -20 -28 -14 14 28 20 7  1
... Reformatted. - _Wolfdieter Lang_, Jan 06 2015
		

Crossrefs

Cf. A113310.

Formula

T(n, k) = C(n-1, n-k) - 2*C(n-2, n-k-1).

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-x + x^3/3!) = -x - 2*x^2/2! - 2*x^3/3! + 5*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014