A117184 Riordan array ((1+x)c(x^2)/sqrt(1-4x^2),xc(x^2)), c(x) the g.f. of A000108.

1, 1, 1, 3, 1, 1, 3, 4, 1, 1, 10, 4, 5, 1, 1, 10, 15, 5, 6, 1, 1, 35, 15, 21, 6, 7, 1, 1, 35, 56, 21, 28, 7, 8, 1, 1, 126, 56, 84, 28, 36, 8, 9, 1, 1, 126, 210, 84, 120, 36, 45, 9, 10, 1, 1, 462, 210, 330, 120, 165, 45, 55, 10, 11, 1, 1

Offset: 0

Author

Paul Barry, Mar 01 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A117186. Diagonal sums are A117187. Inverse is A117185.

Examples

			Triangle begins
1,
1, 1,
3, 1, 1,
3, 4, 1, 1,
10, 4, 5, 1, 1,
10, 15, 5, 6, 1, 1,
35, 15, 21, 6, 7, 1, 1,
35, 56, 21, 28, 7, 8, 1, 1
		

Programs

Mathematica

c[x_] := (1 - Sqrt[1 - 4 x])/(2 x);
(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 + #) c[#^2]/Sqrt[1 - 4 #^2]&, # c[#^2]&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Formula

Number triangle T(n,k)=C(n+1,(n+k)/2+1)(1+(-1)^(n-k))/2+C(n,(n+k)/2+1/2)(1-(-1)^(n-k))/2; Column k has e.g.f. Bessel_I(k,2x)+Bessel_I(k+1,2x)+Bessel_I(k+2,2x).