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%I #17 Nov 05 2020 14:29:19
%S 1,-1,1,1,-3,1,-1,8,-5,1,1,-23,19,-7,1,-1,74,-69,34,-9,1,1,-262,256,
%T -147,53,-11,1,-1,993,-986,615,-265,76,-13,1,1,-3943,3935,-2571,1235,
%U -431,103,-15,1,-1,16178,-16169,10862,-5591,2216,-653,134,-17,1,1
%N Matrix inverse of triangle A063967.
%C Riordan array (1/(1+x), (sqrt(1+6x+5x^2)-x-1)/(2(1+x))). The absolute value array is (1/(1-x),xc(x)/(1-xc(x))) where c(x) is the g.f. of A000108. It factorizes as (1/(1-x),x/(1-x))(1,xc(x)). - _Paul Barry_, Jun 10 2005
%H Lara K. Pudwell, <a href="http://arxiv.org/abs/1408.6823">Ascent sequences and the binomial convolution of Catalan numbers</a>, arXiv preprint arXiv:1408.6823 [math.CO], 2014.
%e From _Paul Barry_, Apr 15 2010: (Start)
%e Triangle begins
%e 1,
%e -1, 1,
%e 1, -3, 1,
%e -1, 8, -5, 1,
%e 1, -23, 19, -7, 1,
%e -1, 74, -69, 34, -9, 1,
%e 1, -262, 256, -147, 53, -11, 1,
%e -1, 993, -986, 615, -265, 76, -13, 1,
%e 1, -3943, 3935, -2571, 1235, -431, 103, -15, 1
%e Production matrix begins
%e -1, 1,
%e 0, -2, 1,
%e 0, 1, -2, 1,
%e 0, -1, 1, -2, 1,
%e 0, 1, -1, 1, -2, 1,
%e 0, -1, 1, -1, 1, -2, 1,
%e 0, 1, -1, 1, -1, 1, -2, 1,
%e 0, -1, 1, -1, 1, -1, 1, -2, 1,
%e 0, 1, -1, 1, -1, 1, -1, 1, -2, 1,
%e 0, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1 (End)
%t rows = 11; t[n_, k_] := Sum[Binomial[j, n - j]*Binomial[j, k], {j, 0, n}]; T = Table[t[n, k], {n, 0, rows - 1}, {k, 0, rows - 1}] // Inverse; Table[ T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Oct 11 2017 *)
%Y Row sums: A091699. Row sums (absolute values): A007317. Column 1: A050511.
%K sign,tabl
%O 0,5
%A _Christian G. Bower_, Jan 29 2004
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