A096470 Triangle T(n,k), read by rows, formed by setting all entries in the zeroth column and in the main diagonal ((n,n) entries) to 1 and defining the rest of the entries by the recursion T(n,k) = T(n-1,k) - T(n,k-1).

1, 1, 1, 1, 0, 1, 1, -1, 2, 1, 1, -2, 4, -3, 1, 1, -3, 7, -10, 11, 1, 1, -4, 11, -21, 32, -31, 1, 1, -5, 16, -37, 69, -100, 101, 1, 1, -6, 22, -59, 128, -228, 329, -328, 1, 1, -7, 29, -88, 216, -444, 773, -1101, 1102, 1, 1, -8, 37, -125, 341, -785, 1558, -2659, 3761, -3760, 1, 1, -9, 46, -171, 512, -1297, 2855, -5514, 9275, -13035, 13036, 1

Offset: 0

Author

Gerald McGarvey, Aug 12 2004

Keywords

• sign
• tabl

Comments

If A(x,y) is the bivariate o.g.f. of a triangular array T(n,k) and B(x,y) is the bivariate o.g.f. of its mirror image T(n,n-k), then B(x,y) = A(x*y, y^(-1)) and A(x,y) = B(x*y, y^(-1)). - Petros Hadjicostas, Aug 08 2020

Examples

			From _Petros Hadjicostas_, Aug 08 2020: (Start)
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  0,  1;
  1, -1,  2,   1;
  1, -2,  4,  -3,   1;
  1, -3,  7, -10,  11,    1;
  1, -4, 11, -21,  32,  -31,   1;
  1, -5, 16, -37,  69, -100, 101,    1;
  1, -6, 22, -59, 128, -228, 329, -328, 1;
  ... (End)
		

Crossrefs

Cf. A000108, A000124, A032357, A033297, A168377.

Programs

PARI

T(n, k) = if ((k==0) || (n==k), 1, if ((n<0) || (k<0), 0, if (n>k, T(n-1, k) - T(n, k-1), 0)));
for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Petros Hadjicostas, Aug 08 2020

Formula

T(n,k) = T(n-1,k) - T(n,k-1) for 1 <= k <= n-1 with T(n,0) = 1 = T(n,n) for n >= 0.

The 2nd column is T(n,2) = A000124(n-2) for n >= 2 (Hogben's central polygonal numbers).

The "first subdiagonal" (unsigned) is |T(n,n-1)| = A032357(n-1) for n >= 1 (Convolution of Catalan numbers and powers of -1).

The "2nd subdiagonal" (unsigned) is |T(n,n-2)| = A033297(n) = Sum_{i=0..n-2} (-1)^i*C(n-1-i) for n >= 2, where C(n) are the Catalan numbers (A000108).

From Petros Hadjicostas, Aug 08 2020: (Start)

|T(n,k)| = |A168377(n,n-k)| for 0 <= k <= n.

Bivariate o.g.f.: (1 + y + x*y*c(-x*y))/((1 - x*y)*(1 - x + y)), where c(x) = 2/(1 + sqrt(1 - 4*x)) = o.g.f. of A000108.

Bivariate o.g.f. of |T(n,k)|: (1 - y - x*y*c(x*y))/((1 + x*y)*(1 - x - y)) + 2*x*y/(1 - x^2*y^2).

Bivariate o.g.f. of mirror image T(n,n-k): (1 + y + x*y*c(-x))/((1 - x)*(1 + y - x*y^2)).

Bivariate o.g.f. of |T(n,n-k)|: (1 - y + x*y*c(x))/((1 + x)*(1 - y + x*y^2)) + 2*x/(1 - x^2). (End)

Extensions

Offset changed to 0 by Petros Hadjicostas, Aug 08 2020