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A103136 Inverse of the Delannoy triangle.

%I A103136 #19 Nov 16 2023 15:56:01
%S A103136 1,-1,1,2,-3,1,-6,10,-5,1,22,-38,22,-7,1,-90,158,-98,38,-9,1,394,-698,
%T A103136 450,-194,58,-11,1,-1806,3218,-2126,978,-334,82,-13,1,8558,-15310,
%U A103136 10286,-4942,1838,-526,110,-15,1,-41586,74614,-50746,25150,-9922,3142,-778,142,-17,1,206098,-370610,254410,-129050
%N A103136 Inverse of the Delannoy triangle.
%C A103136 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)).
%C A103136 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
%C A103136 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
%F A103136 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.
%F A103136 G.f.: 1/(1 + (x - xy)/(1 + x/(1 + 2x/(1 + x/(1 + 2x/(1+... (continued fraction). - _Paul Barry_, Apr 29 2009
%e A103136 From _Paul Barry_, Apr 29 2009: (Start)
%e A103136 Triangle begins
%e A103136     1;
%e A103136    -1,    1;
%e A103136     2,   -3,    1;
%e A103136    -6,   10,   -5,    1;
%e A103136    22,  -38,   22,   -7,    1;
%e A103136   -90,  158,  -98,   38,   -9,    1;
%e A103136   394, -698,  450, -194,   58,  -11,    1;
%e A103136 Production matrix is
%e A103136   -1,  1,
%e A103136    1, -2,  1,
%e A103136   -1,  2, -2,  1,
%e A103136    1, -2,  2, -2,  1,
%e A103136   -1,  2, -2,  2, -2,  1
%e A103136 The unsigned triangle has production matrix
%e A103136   1, 1,
%e A103136   1, 2, 1,
%e A103136   1, 2, 2, 1,
%e A103136   1, 2, 2, 2, 1,
%e A103136   1, 2, 2, 2, 2, 1 (End)
%o A103136 (SageMath)
%o A103136 def A103136(dim): # Returns a triangle with 'dim' rows
%o A103136     M = matrix([[simplify(hypergeometric([-n, n-k], [1], 2))
%o A103136           for n in range(k+1)] + [0]*(dim-k-1) for k in range(dim)])
%o A103136     return [row[:n+1] for n, row in enumerate(M.inverse())]
%o A103136 A103136(9)  # _Peter Luschny_, Nov 16 2023
%K A103136 easy,sign,tabl
%O A103136 0,4
%A A103136 _Paul Barry_, Jan 24 2005