A114193 Riordan array (1/(1+2xc(-2x)),xc(-2x)/(1+2xc(-2x))), c(x) the g.f. of A000108.

1, -2, 1, 8, -6, 1, -40, 36, -10, 1, 224, -224, 80, -14, 1, -1344, 1440, -600, 140, -18, 1, 8448, -9504, 4400, -1232, 216, -22, 1, -54912, 64064, -32032, 10192, -2184, 308, -26, 1, 366080, -439296, 232960, -81536, 20160, -3520, 416, -30, 1, -2489344, 3055104, -1697280, 639744, -176256, 35904, -5304, 540, -34, 1

Offset: 0

Author

Paul Barry, Nov 16 2005

Keywords

• easy
• sign
• tabl

Comments

Row sums are A114191. Diagonal sums are A114194. Inverse of A114192.

Triangle T(n,k), read by rows, given by (-2, -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, Jan 26 2014

Examples

			Triangle begins
      1;
     -2,    1;
      8,   -6,    1;
    -40,   36,  -10,   1;
    224, -224,   80, -14,   1;
  -1344, 1440, -600, 140, -18, 1;
		

Crossrefs

Cf. A039599, A084938.

Programs

Mathematica

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

Formula

Riordan array ((sqrt(1+8x)-1)/(4x), (sqrt(1+8x)-1)^2/(16x)).

T(n, k) = (-2)^(n-k)*A039599(n, k) = (-2)^(n-k)*C(2*n, n-k)*(2*k+1)/(n+k+1). - Philippe Deléham, Nov 17 2005