A106180 Matrix inverse of number triangle A046854.

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

Offset: 0

Author

Paul Barry, Apr 24 2005

Keywords

• easy
• sign
• tabl

Comments

First column is A105523; second column is A106181.

Triangle T(n,k), 0 <= k <= n, read by rows given by [ -1, 1, -1, 1, -1, 1, -1, 1, -1, 1,...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 29 2006

A124448*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 16 2007

Examples

			Triangle begins
   1;
  -1,  1;
   0, -1,  1;
   1, -1, -1,  1;
   0,  2, -2, -1,  1;
  -2,  2,  3, -3, -1,  1;
   0, -5,  5,  4, -4, -1,  1;
		

Crossrefs

Cf. A000108.

Formula

Riordan array (1-y, y) where y=-(1-sqrt(1+4x^2))/(2x).

Sum_{k=0..n} abs(T(n,k)) = A063886(n). - Philippe Deléham, Oct 06 2006

T(0,0)=1; T(n,k)=0 if k < 0 or if k > n; T(n,0) = -T(n-1,0) - T(n-1,1); T(n,k) = T(n,k-1) - T(n-1,k+1) for k >= 1. - Philippe Deléham, Oct 27 2007

T(2n,0) = A000007(n); T(2n+2,2k+2) = -T(2n+2,2k+1) = (-1)^(n-k)*A039598(n,k); T(2n+1,2k+1) = -T(2n+1,2k) = (-1)^(n-k)*A039599(n,k). - Philippe Deléham, Oct 29 2007

Sum_{k>=0} T(m,k)*T(n,k)*(-1)^k = T(m+n,0) = A105523(m+n). - Philippe Deléham, Jan 24 2010