A294034 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.
1, 2, 2, 3, 6, 3, -4, 12, 12, 4, -15, -20, 30, 20, 5, -54, -90, -60, 60, 30, 6, 133, -378, -315, -140, 105, 42, 7, 792, 1064, -1512, -840, -280, 168, 56, 8, 4293, 7128, 4788, -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
Offset: 1
Examples
Triangle starts: [1][ 1] [2][ 2, 2] [3][ 3, 6, 3] [4][ -4, 12, 12, 4] [5][ -15, -20, 30, 20, 5] [6][ -54, -90, -60, 60, 30, 6] [7][ 133, -378, -315, -140, 105, 42, 7] [8][ 792, 1064, -1512, -840, -280, 168, 56, 8]
Programs
-
Maple
gf := exp(x*z)*z*((exp(z) + 1)/((exp(z) + 2*exp(-z/2)*cos(z*sqrt(3)/2))/3) - 1): s := n -> n!*coeff(series(gf, z, 12), z, n): C := n -> PolynomialTools:-CoefficientList(s(n), x): ListTools:-FlattenOnce([seq(C(n), n=1..11)]);