%I #35 Aug 09 2020 01:18:07
%S 1,-1,1,1,0,1,-1,2,1,1,1,3,4,2,1,-1,11,10,7,3,1,1,31,32,21,11,4,1,-1,
%T 101,100,69,37,16,5,1,1,328,329,228,128,59,22,6,1,-1,1102,1101,773,
%U 444,216,88,29,7,1,1,3760,3761,2659,1558,785,341,125,37,8,1
%N Riordan array (1/(1 + x), x*c(x)), where c(x) is the o.g.f. of Catalan numbers A000108.
%H Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, <a href="https://arxiv.org/abs/math/0702638">Production matrices and Riordan arrays</a>, arXiv:math/0702638 [math.CO], 2007.
%H Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, <a href="https://doi.org/10.1007/s00026-009-0013-1">Production matrices and Riordan arrays</a>, Annals of Combinatorics, 13 (2009), 65-85.
%H L. W. Shapiro, S. Getu, W.-J. Woan, and L. C. Woodson, <a href="https://doi.org/10.1016/0166-218X(91)90088-E">The Riordan group</a>, Discrete Applied Mathematics, 34(1-3) (1991), 229-239.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Riordan_array">Riordan array</a>.
%F T(n,0) = (-1)^n and T(n,n) = 1.
%F Sum_{0 <= k <= n} T(n,k) = A032357(n).
%F From _Petros Hadjicostas_, Aug 08 2020: (Start)
%F T(n,k) = T(n,k-1) - T(n-1,k-2) for 2 <= k <= n with initial conditions T(n,0) = (-1)^n (n >= 0) and T(n,1) = A032357(n-1) (n >= 1).
%F T(n,2) = A033297(n).
%F T(n,n-1) = n - 2 for n >= 1.
%F |T(n,k)| = |A096470(n,n-k)| for 0 <= k <= n.
%F Bivariate o.g.f.: 1/((1 + x)*(1 - x*y*c(x))), where c(x) is the o.g.f. of A000108.
%F Bivariate o.g.f.: (1 - y + x*y*c(x))/((1 + x)*(1 - y + x*y^2)).
%F Bivariate o.g.f. of |T(n,k)|: (o.g.f. of T(n,k)) + 2*x/(1 - x^2). (End)
%e Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e 1;
%e -1, 1;
%e 1, 0, 1;
%e -1, 2, 1, 1;
%e 1, 3, 4, 2, 1;
%e -1, 11, 10, 7, 3, 1;
%e 1, 31, 32, 21, 11, 4, 1;
%e -1, 101, 100, 69, 37, 16, 5, 1;
%e ...
%e From _Philippe Deléham_, Sep 14 2014: (Start)
%e Production matrix begins:
%e -1, 1
%e 0, 1, 1
%e 0, 1, 1, 1
%e 0, 1, 1, 1, 1
%e 0, 1, 1, 1, 1, 1
%e 0, 1, 1, 1, 1, 1, 1
%e 0, 1, 1, 1, 1, 1, 1, 1
%e 0, 1, 1, 1, 1, 1, 1, 1, 1
%e ... (End)
%o (PARI) A000108(n) = binomial(2*n, n)/(n+1);
%o A032357(n) = sum(k=0, n, (-1)^(n-k)*A000108(k));
%o T(n, k) = if ((k==0), (-1)^n, if ((n<0) || (k<0), 0, if (k==1, A032357(n-1), if (n > k-1, T(n, k-1) - T(n-1, k-2), 0))));
%o for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Petros Hadjicostas_, Aug 08 2020
%Y Cf. A000012, A000108, A000124, A023443, A032357, A033297, A033999, A091491, A096470, A106566, A127540.
%K sign,tabl
%O 0,8
%A _Philippe Deléham_, Nov 24 2009