A237621 Riordan array (1+x, x*(1-x)); inverse of Riordan array A237619.
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, -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, -5, -14, 20, -8, 1, 0, 0, 0, 0, 0, -1, 9, 0, -28, 27, -9, 1
Offset: 0
Examples
Triangles begins:
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, -1, 2, 5, -5, 1;
0, 0, 0, 0, -3, 0, 9, -6, 1;
0, 0, 0, 0, 1, -5, -5, 14, -7, 1;
...
Production matrix is:
1, 1;
-1, -1, 1;
0, -1, -1, 1;
-1, -2, -1, -1, 1;
-2, -5, -2, -1, -1, 1;
-6, -14, -5, -2, -1, -1, 1;
-18, -42, -14, -5, -2, -1, -1, 1;
-57, -132, -42, -14, -5, -2, -1, -1, 1;
-186, -429, -132, -42, -14, -5, -2, -1, -1, 1;
... (columns are A126983 and A115140)
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
-
Mathematica
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] ]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 26 2022 *) -
SageMath
def T(n,k): # T = A237621 if (k<0 or k>n): return 0 elif (n<2): return 1 else: return T(n-1, k-1) - T(n-2, k-1) flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2022
Formula
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.
Sum_{k=0..n} T(n, k) = A057079(n).