A120616 Generalized Riordan array (1/sqrt(1+4x^2),(1-sqrt(1+4x^2))/(2x)).

1, 0, -1, -2, 0, 1, 0, 3, 0, -1, 6, 0, -4, 0, 1, 0, -10, 0, 5, 0, -1, -20, 0, 15, 0, -6, 0, 1, 0, 35, 0, -21, 0, 7, 0, -1, 70, 0, -56, 0, 28, 0, -8, 0, 1, 0, -126, 0, 84, 0, -36, 0, 9, 0, -1, -252, 0, 210, 0, -120, 0, 45, 0, -10, 0, 1

Offset: 0

Author

Paul Barry, Jun 17 2006

Keywords

• easy
• sign
• tabl
• changed

Comments

Product by A007318 is A104505.

Examples

			Triangle begins
     1;
     0,   -1;
    -2,    0,   1;
     0,    3,   0,  -1;
     6,    0,  -4,   0,    1;
     0,  -10,   0,   5,    0,  -1;
   -20,    0,  15,   0,   -6,   0,  1;
     0,   35,   0, -21,    0,   7,  0, -1;
    70,    0, -56,   0,   28,   0, -8,  0,   1;
     0, -126,   0,  84,    0, -36,  0,  9,   0, -1;
  -252,    0, 210,   0, -120,   0, 45,  0, -10,  0, 1;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened.)
• Robert S. Maier, Sheffer Polynomials and the s-ordering of Exponential Boson Operators, arXiv:2508.13094 [quant-ph], 2025. See p. 27.

Programs

Mathematica

T[n_, k_] := Binomial[n, (n + k)/2]*(-1)^((n + k)/2) (1 + (-1)^(n + k))/2; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 22 2025 *)

Formula

Number triangle T(n,k)=C(n,(n+k)/2)(-1)^((n+k)/2)(1+(-1)^(n+k))/2.

abs(T(n,k)) = A108044(n,k).