A114189 Riordan array (1/(1+xc(-2x)), xc(-2x)/(1+xc(-2x))), c(x) the g.f. of A000108.

1, -1, 1, 3, -4, 1, -13, 19, -7, 1, 67, -102, 44, -10, 1, -381, 593, -278, 78, -13, 1, 2307, -3640, 1795, -568, 121, -16, 1, -14589, 23231, -11849, 4051, -999, 173, -19, 1, 95235, -152650, 79750, -28770, 7820, -1598, 234, -22, 1, -636925, 1025965, -545680, 204760, -59650, 13642, -2392, 304, -25, 1

Offset: 0

Author

Paul Barry, Nov 16 2005

Keywords

• easy
• sign
• tabl

Comments

Inverse of A114188. Factors as (1,xc(-2x))*(1/(1+x), x/(1+x)). Row sums are 0^n. Diagonal sums are A114190. First column is A114191. A signed version of A110506.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -1,-2,-2,-2,-2,-2,-2,...] DELTA [1,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 01 2007

Examples

			Triangle begins
     1;
    -1,    1;
     3,   -4,    1;
   -13,   19,   -7,   1;
    67, -102,   44, -10,   1;
  -381,  593, -278,  78, -13, 1;
		

Programs

Mathematica

c[x_] := (1 - Sqrt[1 - 4x])/(2x);
(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1 + # c[-2#])&, # c[-2#]/(1 + # c[-2#])&, 10] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Formula

Riordan array ((3-sqrt(1+8x))/(2(1-x)), (sqrt(1+8x)-2x-1)/(2(1-x))).

T(n,k) = (-1)^(n-k)*A110506(n,k). - Philippe Deléham, Mar 24 2007