%I #6 Oct 31 2017 14:33:33
%S 1,2,2,3,6,3,-4,12,12,4,-15,-20,30,20,5,-54,-90,-60,60,30,6,133,-378,
%T -315,-140,105,42,7,792,1064,-1512,-840,-280,168,56,8,4293,7128,4788,
%U -4536,-1890,-504,252,72,9,-15130,42930,35640,15960,-11340,-3780,-840,360,90,10,-123849,-166430,236115,130680,43890,-24948,-6930,-1320,495,110,11
%N Triangle read by rows, expansion of exp(x*z)*z*((exp(z) + 1)/((exp(z) + 2*exp(-z/2)*cos(z*sqrt(3)/2))/3) -1), for n >= 1 and 0 <= k <= n-1.
%e Triangle starts:
%e [1][ 1]
%e [2][ 2, 2]
%e [3][ 3, 6, 3]
%e [4][ -4, 12, 12, 4]
%e [5][ -15, -20, 30, 20, 5]
%e [6][ -54, -90, -60, 60, 30, 6]
%e [7][ 133, -378, -315, -140, 105, 42, 7]
%e [8][ 792, 1064, -1512, -840, -280, 168, 56, 8]
%p gf := exp(x*z)*z*((exp(z) + 1)/((exp(z) + 2*exp(-z/2)*cos(z*sqrt(3)/2))/3) - 1):
%p s := n -> n!*coeff(series(gf, z, 12), z, n):
%p C := n -> PolynomialTools:-CoefficientList(s(n), x):
%p ListTools:-FlattenOnce([seq(C(n), n=1..11)]);
%Y Cf. A003506, A294033.
%K sign,tabl
%O 1,2
%A _Peter Luschny_, Oct 24 2017