A168377 Riordan array (1/(1 + x), x*c(x)), where c(x) is the o.g.f. of Catalan numbers A000108.
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
Offset: 0
Examples
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)
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.
Crossrefs
Programs
-
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
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)