A103136 Inverse of the Delannoy triangle.

1, -1, 1, 2, -3, 1, -6, 10, -5, 1, 22, -38, 22, -7, 1, -90, 158, -98, 38, -9, 1, 394, -698, 450, -194, 58, -11, 1, -1806, 3218, -2126, 978, -334, 82, -13, 1, 8558, -15310, 10286, -4942, 1838, -526, 110, -15, 1, -41586, 74614, -50746, 25150, -9922, 3142, -778, 142, -17, 1, 206098, -370610, 254410, -129050

Offset: 0

Author

Paul Barry, Jan 24 2005

Keywords

• easy
• sign
• tabl

Comments

The Delannoy triangle is A008288 viewed as a number triangle. It is then given by the Riordan array (1/(1-x), x(1+x)/(1-x)). The absolute value of A103136 is the Riordan array (1+xS(x),xS(x)) which is the inverse of the signed Delannoy triangle (1/(1+x), x(1-x)/(1+x)).

Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -1, -1, -2, -1, -2, -1, -2, -1, -2, ... ] DELTA [ 1, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938; the unsigned version is given by [ 1, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA [ 1, 0, 0, 0, 0, 0, 0, 0, ... ]. - Philippe Deléham, Jul 08 2005

The unsigned number |T(n,k)| counts Schroeder n-paths whose ascent starting at the initial vertex has length k. A Schroeder n-path is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (upsteps), D=(1,-1) (downsteps) and F=(2,0) (flatsteps) and never going below the x-axis. For example, |T(2,0)| = 2 counts FF, FUD; |T(2,1)| = 3 counts UFD, UDF, UDUD; |T(2,2)| = 1 counts UUDD. - David Callan, Jul 14 2006

Examples

			From _Paul Barry_, Apr 29 2009: (Start)
Triangle begins
    1;
   -1,    1;
    2,   -3,    1;
   -6,   10,   -5,    1;
   22,  -38,   22,   -7,    1;
  -90,  158,  -98,   38,   -9,    1;
  394, -698,  450, -194,   58,  -11,    1;
Production matrix is
  -1,  1,
   1, -2,  1,
  -1,  2, -2,  1,
   1, -2,  2, -2,  1,
  -1,  2, -2,  2, -2,  1
The unsigned triangle has production matrix
  1, 1,
  1, 2, 1,
  1, 2, 2, 1,
  1, 2, 2, 2, 1,
  1, 2, 2, 2, 2, 1 (End)
		

Programs

SageMath

def A103136(dim): # Returns a triangle with 'dim' rows
    M = matrix([[simplify(hypergeometric([-n, n-k], [1], 2))
          for n in range(k+1)] + [0]*(dim-k-1) for k in range(dim)])
    return [row[:n+1] for n, row in enumerate(M.inverse())]
A103136(9)  # Peter Luschny, Nov 16 2023

Formula

Riordan array (1-f(x), f(x)) with f(x) = xS(-x), S(x) the g.f. of the large Schroeder numbers A006318. Equivalent to Riordan array (g(x), 1-g(x)) where g(x) = (3+x-sqrt(1+6x+x^2))/2.

G.f.: 1/(1 + (x - xy)/(1 + x/(1 + 2x/(1 + x/(1 + 2x/(1+... (continued fraction). - Paul Barry, Apr 29 2009