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%I #11 May 27 2022 08:11:15
%S 1,1,1,0,0,1,0,-1,-1,1,0,0,-1,-2,1,0,0,1,0,-3,1,0,0,0,2,2,-4,1,0,0,0,
%T -1,2,5,-5,1,0,0,0,0,-3,0,9,-6,1,0,0,0,0,1,-5,-5,14,-7,1,0,0,0,0,0,4,
%U -5,-14,20,-8,1,0,0,0,0,0,-1,9,0,-28,27,-9,1
%N Riordan array (1+x, x*(1-x)); inverse of Riordan array A237619.
%H G. C. Greubel, <a href="/A237621/b237621.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n,k) = T(n-1,k-1) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
%F Sum_{k=0..n} T(n, k) = A057079(n).
%e Triangles begins:
%e 1;
%e 1, 1;
%e 0, 0, 1;
%e 0, -1, -1, 1;
%e 0, 0, -1, -2, 1;
%e 0, 0, 1, 0, -3, 1;
%e 0, 0, 0, 2, 2, -4, 1;
%e 0, 0, 0, -1, 2, 5, -5, 1;
%e 0, 0, 0, 0, -3, 0, 9, -6, 1;
%e 0, 0, 0, 0, 1, -5, -5, 14, -7, 1;
%e ...
%e Production matrix is:
%e 1, 1;
%e -1, -1, 1;
%e 0, -1, -1, 1;
%e -1, -2, -1, -1, 1;
%e -2, -5, -2, -1, -1, 1;
%e -6, -14, -5, -2, -1, -1, 1;
%e -18, -42, -14, -5, -2, -1, -1, 1;
%e -57, -132, -42, -14, -5, -2, -1, -1, 1;
%e -186, -429, -132, -42, -14, -5, -2, -1, -1, 1;
%e ... (columns are A126983 and A115140)
%t T[n_, k_]:= T[n,k]= If[k<0 || k>n, 0, If[n<2, 1, T[n-1,k-1] - T[n-2,k-1] ]];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 26 2022 *)
%o (SageMath)
%o def T(n,k): # T = A237621
%o if (k<0 or k>n): return 0
%o elif (n<2): return 1
%o else: return T(n-1, k-1) - T(n-2, k-1)
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 26 2022
%Y Cf. A057079 (row sums), A237619.
%K sign,tabl
%O 0,14
%A _Philippe Deléham_, Feb 10 2014
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