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

0,4

Comments

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

Links

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

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).

Example

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

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 *)

Crossrefs

Sequence in context: A051120 A114476 A260419 * A035690 A124794 A206496

Adjacent sequences: A117181 A117182 A117183 * A117185 A117186 A117187

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 01 2006

Status

approved