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

0,9

Links

Table of n, a(n) for n=0..65.

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

Formula

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

Example

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;
  ...

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

Crossrefs

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

Triangle A154926 with 1st column A000007.

Sequence in context: A119337 A213889 A363779 * A110555 A097805 A071919

Adjacent sequences: A373161 A373162 A373163 * A373165 A373166 A373167

Keyword

sign,tabl

Author

Stefano Spezia, May 26 2024

Status

approved