A186827 Riordan array (1-x, x(1-x)/(1+x)).

1, -1, 1, 0, -3, 1, 0, 4, -5, 1, 0, -4, 12, -7, 1, 0, 4, -20, 24, -9, 1, 0, -4, 28, -56, 40, -11, 1, 0, 4, -36, 104, -120, 60, -13, 1, 0, -4, 44, -168, 280, -220, 84, -15, 1, 0, 4, -52, 248, -552, 620, -364, 112, -17, 1, 0, -4, 60, -344, 968, -1452, 1204, -560, 144, -19, 1

Offset: 0

Author

Paul Barry, Feb 27 2011

Keywords

• sign
• easy
• tabl

Comments

Inverse of A186826. Row sums are A176742. Diagonal sums are the alternating sign tribonacci numbers (-1)^n*A000213(n).

Examples

			Triangle begins
1,
-1, 1,
0, -3, 1,
0, 4, -5, 1,
0, -4, 12, -7, 1,
0, 4, -20, 24, -9, 1,
0, -4, 28, -56, 40, -11, 1,
0, 4, -36, 104, -120, 60, -13, 1,
0, -4, 44, -168, 280, -220, 84, -15, 1,
0, 4, -52, 248, -552, 620, -364, 112, -17, 1,
0, -4, 60, -344, 968, -1452, 1204, -560, 144, -19, 1
		

References

• C.-P. Chou, H. A. Witek, ZZDecomposer: A Graphical Toolkit for Analyzing the Zhang-Zhang Polynomials of Benzenoid Structures, MATCH: Communications in Mathematical and in Computer Chemistry. 71 (2014) 741-764. See Eq. (13). - N. J. A. Sloane, Jul 03 2014

Formula

Triangle T(n,k)=(-1)^(n-k)*sum{j=0..k+1, binomial(k+1,j)*binomial(n-j-1,n-k-j)}.

T(n,k)=T(n-1,k-1)-T(n-1,k)-T(n-2,k-1), T(0,0)=1, T(1,0)=-1, T(1,1)=1, T(2,0)=0, T(2,1)=-3, T(2,2)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 12 2014