A373164 Triangle read by rows: the exponential almost-Riordan array ( 1 | 2 - exp(x), x ).

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, -1, -3, -3, 1, 0, -1, -4, -6, -4, 1, 0, -1, -5, -10, -10, -5, 1, 0, -1, -6, -15, -20, -15, -6, 1, 0, -1, -7, -21, -35, -35, -21, -7, 1, 0, -1, -8, -28, -56, -70, -56, -28, -8, 1, 0, -1, -9, -36, -84, -126, -126, -84, -36, -9, 1

Offset: 0

Author

Stefano Spezia, May 26 2024

Keywords

• sign
• tabl

Examples

			The triangle begins:
  1;
  0,  1;
  0, -1,  1;
  0, -1, -2,   1;
  0, -1, -3,  -3,   1;
  0, -1, -4,  -6,  -4,   1;
  0, -1, -5, -10, -10,  -5,   1;
  0, -1, -6, -15, -20, -15,  -6,  1;
  0, -1, -7, -21, -35, -35, -21, -7, 1;
  ...
		

Links

• Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 8.

Crossrefs

Cf. A000012 (right diagonal), A024000 (subdiagonal), A122958 (row sums), A153881 (k=1).

Triangle A154926 with 1st column A000007.

Programs

Mathematica

T[n_,0]:=KroneckerDelta[n,0]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[(2-Exp[x])x^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

Formula

T(n,0) = A000007(n); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] (2-exp(x))*x^(k-1).