A215060 Triangle read by rows, e.g.f. exp(x*(z+1/2))/((exp(3*x/2) + 2*cos(sqrt(3)*x/2))/3).

1, 0, 1, 0, 0, 1, -1, 0, 0, 1, 0, -4, 0, 0, 1, 0, 0, -10, 0, 0, 1, 19, 0, 0, -20, 0, 0, 1, 0, 133, 0, 0, -35, 0, 0, 1, 0, 0, 532, 0, 0, -56, 0, 0, 1, -1513, 0, 0, 1596, 0, 0, -84, 0, 0, 1, 0, -15130, 0, 0, 3990, 0, 0, -120, 0, 0, 1, 0, 0, -83215, 0

Offset: 0

Author

Peter Luschny, Aug 01 2012

Keywords

• sign
• tabl

Examples

			[0] [1]
[1] [0, 1]
[2] [0, 0, 1]
[3] [-1, 0, 0, 1]
[4] [0, -4, 0, 0, 1]
[5] [0, 0, -10, 0, 0, 1]
[6] [19, 0, 0, -20, 0, 0, 1]
[7] [0, 133, 0, 0, -35, 0, 0, 1]
[8] [0, 0, 532, 0, 0, -56, 0, 0, 1]
[9] [-1513, 0, 0, 1596, 0, 0, -84, 0, 0, 1]
		

Crossrefs

Cf. A215061, A215062, A215063, A215064, A215065.

Programs

Sage

def triangle(f, dim):
    var('x,z')
    s = f.series(x, dim+2)
    P = [factorial(i)*s.coefficient(x,i) for i in range(dim)]
    for k in range(dim): print([k], [P[k].coefficient(z,i) for i in (0..k)])
def A215060_triangle(dim) :
    var('x, z')
    f = exp(x*(z+1/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)
    return triangle(f, dim)
A215060_triangle(12)

Formula

Matrix inverse is A215061.

T(n,k) = A215064(n,k) - A215062(n,k) + [n==k].

|T(3*n,0)| = A002115(n).