A098435 - OEIS

%I #18 Jan 22 2020 21:09:01

%S 1,-1,1,2,-3,1,-8,13,-6,1,56,-92,45,-10,1,-608,1000,-493,115,-15,1,

%T 9440,-15528,7662,-1799,245,-21,1,-198272,326144,-160944,37817,-5180,

%U 462,-28,1,5410688,-8900224,4392080,-1032088,141465,-12684,798,-36,1

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

%C Inverse matrix of A054142. - _Paul Barry_, Jan 21 2005

%C 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

%H D. Dumont and J. Zeng, <a href="http://math.univ-lyon1.fr/homes-www/zeng/public_html/paper/publication.html">Polynomes d'Euler et les fractions continues de Stieltjes-Rogers</a>, Ramanujan J. 2 (1998) 3, 387-410.

%F See A065547 for formulas.

%e    1;

%e   -1,   1;

%e    2,  -3,   1;

%e   -8,  13,  -6,   1;

%e   56, -92,  45, -10,  1;

%t 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_ *)

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

%K tabl,sign

%O 1,4

%A _Ralf Stephan_, Sep 08 2004