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%I #33 Feb 18 2020 11:37:38
%S 1,2,-1,3,-4,1,4,-11,6,-1,5,-26,23,-8,1,6,-57,72,-39,10,-1,7,-120,201,
%T -150,59,-12,1,8,-247,522,-501,268,-83,14,-1,9,-502,1291,-1524,1037,
%U -434,111,-16,1,10,-1013,3084,-4339,3598,-1905,656,-143,18,-1
%N Triangle read by rows: T(n,0) = n+1, T(n,k) = 2*T(n-1,k) - T(n-1,k-1), T(n,k) = 0 if k > n and if k < 0.
%C A Riordan array - see the Luzon references.
%C The second column is A000295 signed. - _Michel Marcus_, Feb 14 2014
%H Ana Luzón, <a href="http://arxiv.org/abs/0907.2328">Iterative Processes Related to Riordan Arrays: The Reciprocation and the Inversion of Power Series</a>, arXiv:0907.2328 [math.CO]; Discrete Math., 310 (2010), 3607-3618.
%H Ana Luzón and Manuel A. Morón, <a href="http://dx.doi.org/10.1016/j.laa.2008.12.001">Riordan matrices in the reciprocation of quadratic polynomials</a>, Linear Algebra Appl. 430 (2009), no. 8-9, 22542270.
%F From _R. J. Mathar_, May 31 2009: (Start)
%F Sum_{k=0..n} T(n,k) = A080956(n).
%F Conjecture: Sum_{i=0..n} |T(n,k)| = A047926(n). (End)
%F T(n,k) = (-1)^k*Sum_{i=0..n-k} binomial(n+1,i+k+1)*binomial(i+k-1,k). - _Vladimir Kruchinin_, Nov 22 2016
%F G.f.: (1-2*x)/(1-x)^2/(1-2*x+y*x). - _Vladimir Kruchinin_, Nov 22 2016
%e Triangle begins
%e 1;
%e 2, -1;
%e 3, -4, 1;
%e 4, -11, 6, -1;
%e 5, -26, 23, -8, 1;
%e 6, -57, 72, -39, 10, -1;
%e 7, -120, 201, -150, 59, -12, 1;
%e ...
%t With[{m = 9}, CoefficientList[CoefficientList[Series[(1-2*x)/(1-x)^2/(1-2*x
%t +y*x), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* _Georg Fischer_, Feb 18 2020 *)
%o (Maxima)
%o T(n,k):=coeff(taylor(1/(1-x)^2*(-x/(1-x))^k,x,0,15),x,n); /* _Vladimir Kruchinin_, Nov 22 2016 */
%Y Cf. A000295, A047926, A080956, A181690.
%K easy,sign,tabl
%O 0,2
%A _Philippe Deléham_, Apr 24 2009
%E a(41) corrected by _Georg Fischer_, Feb 18 2020
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