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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).

%I A215064 #17 Mar 23 2020 17:32:14
%S A215064 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,
%T A215064 6,1,99,133,-189,-105,-35,21,7,1,477,792,532,-504,-210,-56,28,8,1,
%U A215064 -1513,4293,3564,1596,-1134,-378,-84,36,9,1,-11259
%N 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).
%F A215064 Matrix inverse is A215065.
%F A215064 T(n,k) = A215060(n,k) + A215062(n,k) - [n==k].
%F A215064 |T(n,0)| = A178963(n).
%F A215064 |T(3*n,0)| = A002115(n).
%e A215064 [0] [1]
%e A215064 [1] [1, 1]
%e A215064 [2] [1, 2, 1]
%e A215064 [3] [-1, 3, 3, 1]
%e A215064 [4] [-3, -4, 6, 4, 1]
%e A215064 [5] [-9, -15, -10, 10, 5, 1]
%e A215064 [6] [19, -54, -45, -20, 15, 6, 1]
%e A215064 [7] [99, 133, -189, -105, -35, 21, 7, 1]
%e A215064 [8] [477, 792, 532, -504, -210, -56, 28, 8, 1]
%e A215064 [9] [-1513, 4293, 3564, 1596, -1134, -378, -84, 36, 9, 1]
%t A215064 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 *)
%o A215064 (Sage) # uses[triangle from A215060]
%o A215064 def A215064_triangle(dim):
%o A215064     var('x, z')
%o A215064     f = exp(x*z)*((exp(x/2)+exp(x*3/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)-1)
%o A215064     return triangle(f, dim)
%o A215064 A215064_triangle(12)
%Y A215064 Cf. A215060, A215061, A215062, A215063, A215065.
%K A215064 sign,tabl
%O A215064 0,5
%A A215064 _Peter Luschny_, Aug 01 2012