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A237621 Riordan array (1+x, x*(1-x)); inverse of Riordan array A237619.

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