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A159856 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.

%I A159856 #37 May 09 2024 15:55:11
%S A159856 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 A159856 -150,59,-12,1,8,-247,522,-501,268,-83,14,-1,9,-502,1291,-1524,1037,
%U A159856 -434,111,-16,1,10,-1013,3084,-4339,3598,-1905,656,-143,18,-1
%N A159856 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 A159856 A Riordan array - see the Luzon references.
%C A159856 The second column is A000295 signed. - _Michel Marcus_, Feb 14 2014
%H A159856 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 A159856 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 A159856 From _R. J. Mathar_, May 31 2009: (Start)
%F A159856 Sum_{k=0..n} T(n,k) = A080956(n).
%F A159856 Conjecture: Sum_{i=0..n} |T(n,k)| = A047926(n). (End)
%F A159856 T(n,k) = (-1)^k*Sum_{i=0..n-k} binomial(n+1,i+k+1)*binomial(i+k-1,k-1) for k > 0. - _Vladimir Kruchinin_, Nov 22 2016 [corrected by _Werner Schulte_, May 09 2024]
%F A159856 G.f.: (1-2*x)/(1-x)^2/(1-2*x+y*x). - _Vladimir Kruchinin_, Nov 22 2016
%e A159856 Triangle begins
%e A159856   1;
%e A159856   2,   -1;
%e A159856   3,   -4,    1;
%e A159856   4,  -11,    6,   -1;
%e A159856   5,  -26,   23,   -8,    1;
%e A159856   6,  -57,   72,  -39,   10,   -1;
%e A159856   7, -120,  201, -150,   59,  -12,    1;
%e A159856   ...
%t A159856 With[{m = 9}, CoefficientList[CoefficientList[Series[(1-2*x)/(1-x)^2/(1-2*x
%t A159856 +y*x), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* _Georg Fischer_, Feb 18 2020 *)
%o A159856 (Maxima)
%o A159856 T(n,k):=coeff(taylor(1/(1-x)^2*(-x/(1-x))^k,x,0,15),x,n); /* _Vladimir Kruchinin_, Nov 22 2016 */
%Y A159856 Cf. A000295, A047926, A080956, A181690.
%K A159856 easy,sign,tabl
%O A159856 0,2
%A A159856 _Philippe Deléham_, Apr 24 2009
%E A159856 a(41) corrected by _Georg Fischer_, Feb 18 2020