A154556 Exponential Riordan array [exp(-x), x(1+x/2)].

1, -1, 1, 1, -1, 1, -1, 0, 0, 1, 1, 2, -3, 2, 1, -1, -5, 5, -5, 5, 1, 1, 9, 0, -5, 0, 9, 1, -1, -14, -21, 35, -35, 21, 14, 1, 1, 20, 70, -56, 35, -56, 70, 20, 1, -1, -27, -162, -42, 189, -189, 42, 162, 27, 1

Offset: 0

Author

Paul Barry, Jan 11 2009

Keywords

• easy
• sign
• tabl

Comments

Row sums are double factorials A001147 (aerated).

Product A007318*(this triangle) is A122848.

Examples

			Triangle begins
1,
-1, 1,
1, -1, 1,
-1, 0, 0, 1,
1, 2, -3, 2, 1,
-1, -5, 5, -5, 5, 1,
1, 9, 0, -5, 0, 9, 1,
-1, -14, -21, 35, -35, 21, 14, 1,
1, 20, 70, -56, 35, -56, 70, 20, 1,
-1, -27, -162, -42, 189, -189, 42, 162, 27, 1,
1, 35, 315, 510, -735, 693, -735, 510, 315, 35, 1
		

Crossrefs

Cf. A001147, A007318, A122848.

Programs

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[E^-#&, # (1 + #/2)&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Formula

T(n,k) = (n!/k!)*sum{j=0..k, (-1)^(n-k)*C(k,j)*(-1/2)^j/(n-k-j)!};

T(n,k) = sum{j=0..n, (-1)^(n-j)*C(n,j)*C(j,k)*k!/((2k-j)!*2^(j-k))};