A166353 Exponential Riordan array [1+x*tan(x/2),x].

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 0, 6, 0, 1, 0, 5, 0, 10, 0, 1, 3, 0, 15, 0, 15, 0, 1, 0, 21, 0, 35, 0, 21, 0, 1, 17, 0, 84, 0, 70, 0, 28, 0, 1, 0, 153, 0, 252, 0, 126, 0, 36, 0, 1, 155, 0, 765, 0, 630, 0, 210, 0, 45, 0, 1

Offset: 0

Author

Paul Barry, Oct 12 2009

Keywords

• easy
• nonn
• tabl

Comments

First column is aerated Genocchi number variant with e.g.f. 1+x*tan(x/2).

Row sums are A166354. Diagonal sums are A166355.

Examples

			Triangle begins
1,
0, 1,
1, 0, 1,
0, 3, 0, 1,
1, 0, 6, 0, 1,
0, 5, 0, 10, 0, 1,
3, 0, 15, 0, 15, 0, 1,
0, 21, 0, 35, 0, 21, 0, 1,
17, 0, 84, 0, 70, 0, 28, 0, 1,
0, 153, 0, 252, 0, 126, 0, 36, 0, 1,
155, 0, 765, 0, 630, 0, 210, 0, 45, 0, 1
		

Crossrefs

Cf. A110501, A166354, A166355.

Programs

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1 + # Tan[#/2]&, #&, 11, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Formula

T(n,k) = [k<=n]*G((n-k)/2)*C(n,k)*(1+(-1)^(n-k))/2 where

G(n)=0^n+2(-1)^n*(1-4^n)*sum{k=0..2n, sum{j=0..k, (-1)^j*C(k,j)*j^(2n)/(k+1)}}.