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 A168377 Riordan array (1/(1 + x), x*c(x)), where c(x) is the o.g.f. of Catalan numbers A000108. 2
 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, 101, 100, 69, 37, 16, 5, 1, 1, 328, 329, 228, 128, 59, 22, 6, 1, -1, 1102, 1101, 773, 444, 216, 88, 29, 7, 1, 1, 3760, 3761, 2659, 1558, 785, 341, 125, 37, 8, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007. Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production matrices and Riordan arrays, Annals of Combinatorics, 13 (2009), 65-85. L. W. Shapiro, S. Getu, W.-J. Woan, and L. C. Woodson, The Riordan group, Discrete Applied Mathematics, 34(1-3) (1991), 229-239. Wikipedia, Riordan array. FORMULA T(n,0) = (-1)^n and T(n,n) = 1. Sum_{0 <= k <= n} T(n,k) = A032357(n). From Petros Hadjicostas, Aug 08 2020: (Start) 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). T(n,2) = A033297(n). T(n,n-1) = n - 2 for n >= 1. |T(n,k)| = |A096470(n,n-k)| for 0 <= k <= n. Bivariate o.g.f.: 1/((1 + x)*(1 - x*y*c(x))), where c(x) is the o.g.f. of A000108. Bivariate o.g.f.: (1 - y + x*y*c(x))/((1 + x)*(1 - y + x*y^2)). Bivariate o.g.f. of |T(n,k)|: (o.g.f. of T(n,k)) + 2*x/(1 - x^2). (End) EXAMPLE Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:    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, 101, 100, 69, 37, 16, 5, 1;   ... From Philippe Deléham, Sep 14 2014: (Start) Production matrix begins:   -1, 1    0, 1, 1    0, 1, 1, 1    0, 1, 1, 1, 1    0, 1, 1, 1, 1, 1    0, 1, 1, 1, 1, 1, 1    0, 1, 1, 1, 1, 1, 1, 1    0, 1, 1, 1, 1, 1, 1, 1, 1    ... (End) PROG (PARI) A000108(n) = binomial(2*n, n)/(n+1); A032357(n) = sum(k=0, n, (-1)^(n-k)*A000108(k)); 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)))); for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Petros Hadjicostas, Aug 08 2020 CROSSREFS Cf. A000012, A000108, A000124, A023443, A032357, A033297, A033999, A091491, A096470, A106566, A127540. Sequence in context: A293991 A288638 A261494 * A122867 A124775 A140075 Adjacent sequences:  A168374 A168375 A168376 * A168378 A168379 A168380 KEYWORD sign,tabl AUTHOR Philippe Deléham, Nov 24 2009 STATUS approved

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Last modified April 10 22:17 EDT 2021. Contains 342856 sequences. (Running on oeis4.)