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

0,8

Links

Table of n, a(n) for n=0..65.

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: A261494 A365673 A349574 * A122867 A124775 A140075

Adjacent sequences: A168374 A168375 A168376 * A168378 A168379 A168380

Keyword

sign,tabl

Author

Philippe Deléham, Nov 24 2009

Status

approved