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