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

0,5

Comments

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

References

C.-P. Chou and 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

Links

Table of n, a(n) for n=0..65.

Jorge Ballarín, Jorge Delgado, and Juan Manuel Peña, Accurate computations with Riordan arrays associated with Schröder matrices, Calcolo 63 (2026), Art. 13. See p. 10 (Prop. 3, Ex. 4).

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

Example

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

Crossrefs

Sequence in context: A292506 A212186 A274662 * A207327 A319083 A378154

Adjacent sequences: A186824 A186825 A186826 * A186828 A186829 A186830

Keyword

sign,easy,tabl

Author

Paul Barry, Feb 27 2011

Status

approved