A098435 Triangle of Salie numbers T(n,k) for negative n,k, n < k.

1, -1, 1, 2, -3, 1, -8, 13, -6, 1, 56, -92, 45, -10, 1, -608, 1000, -493, 115, -15, 1, 9440, -15528, 7662, -1799, 245, -21, 1, -198272, 326144, -160944, 37817, -5180, 462, -28, 1, 5410688, -8900224, 4392080, -1032088, 141465, -12684, 798, -36, 1

Offset: 1

Author

Ralf Stephan, Sep 08 2004

Keywords

• tabl
• sign

Comments

Inverse matrix of A054142. - Paul Barry, Jan 21 2005

Essentially the same as the triangle giving by [0,-1,-1,-4,-4,-9,-9,-16,-16,-25,...] DELTA[1,0,1,0,1,0,1,0,1,0,...] = 1; 0,1; 0,-1,1; 0,2,-3,1; 0,-8,13,-6,1; 0,56,-92,45,-10,1; ... where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 30 2006

Examples

			   1;
  -1,   1;
   2,  -3,   1;
  -8,  13,  -6,   1;
  56, -92,  45, -10,  1;
		

Links

• D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.

Crossrefs

T(-1, k) = (-1)^k*A005439(k-1). Row sums are zero.

Programs

Mathematica

rows = 9; A054142 = Table[ PadRight[ Table[ Binomial[2*n-k, k], {k, 0, n}], rows], {n, 0, rows-1}]; inv = Inverse[A054142]; Table[ Take[inv[[n]], n], {n, 1, rows}] // Flatten (* Jean-François Alcover, Oct 02 2013, after Paul Barry *)

Formula

See A065547 for formulas.