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

0,14

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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).

Example

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)

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 *)

Prog

(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

Crossrefs

Cf. A057079 (row sums), A237619.

Sequence in context: A345415 A096608 A155102 * A214625 A363339 A334195

Adjacent sequences: A237618 A237619 A237620 * A237622 A237623 A237624

Keyword

sign,tabl

Author

Philippe Deléham, Feb 10 2014

Status

approved