A215064 Triangle read by rows, e.g.f. exp(x*z)*((exp(x/2)+exp(x*3/2))/((exp(3*x/2)+ 2*cos(sqrt(3)*x/2))/3)-1).

1, 1, 1, 1, 2, 1, -1, 3, 3, 1, -3, -4, 6, 4, 1, -9, -15, -10, 10, 5, 1, 19, -54, -45, -20, 15, 6, 1, 99, 133, -189, -105, -35, 21, 7, 1, 477, 792, 532, -504, -210, -56, 28, 8, 1, -1513, 4293, 3564, 1596, -1134, -378, -84, 36, 9, 1, -11259

Offset: 0

Author

Peter Luschny, Aug 01 2012

Keywords

• sign
• tabl

Examples

			[0] [1]
[1] [1, 1]
[2] [1, 2, 1]
[3] [-1, 3, 3, 1]
[4] [-3, -4, 6, 4, 1]
[5] [-9, -15, -10, 10, 5, 1]
[6] [19, -54, -45, -20, 15, 6, 1]
[7] [99, 133, -189, -105, -35, 21, 7, 1]
[8] [477, 792, 532, -504, -210, -56, 28, 8, 1]
[9] [-1513, 4293, 3564, 1596, -1134, -378, -84, 36, 9, 1]
		

Crossrefs

Cf. A215060, A215061, A215062, A215063, A215065.

Programs

Mathematica

max = 11; f = Exp[x*z]*((Exp[x/2] + Exp[x*(3/2)])/((Exp[3*(x/2)] + 2*Cos[Sqrt[3]*(x/2)])/3) - 1); coes = CoefficientList[ Series[f, {x, 0, max}, {z, 0, max}], {x, z}]; Table[ coes[[n, k]]*(n - 1)!, {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

Sage

# uses[triangle from A215060]
def A215064_triangle(dim):
    var('x, z')
    f = exp(x*z)*((exp(x/2)+exp(x*3/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)-1)
    return triangle(f, dim)
A215064_triangle(12)

Formula

Matrix inverse is A215065.

T(n,k) = A215060(n,k) + A215062(n,k) - [n==k].

|T(n,0)| = A178963(n).

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