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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.

%I A294034 #6 Oct 31 2017 14:33:33
%S A294034 1,2,2,3,6,3,-4,12,12,4,-15,-20,30,20,5,-54,-90,-60,60,30,6,133,-378,
%T A294034 -315,-140,105,42,7,792,1064,-1512,-840,-280,168,56,8,4293,7128,4788,
%U A294034 -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 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.
%e A294034 Triangle starts:
%e A294034 [1][   1]
%e A294034 [2][   2,    2]
%e A294034 [3][   3,    6,     3]
%e A294034 [4][  -4,   12,    12,    4]
%e A294034 [5][ -15,  -20,    30,   20,    5]
%e A294034 [6][ -54,  -90,   -60,   60,   30,   6]
%e A294034 [7][ 133, -378,  -315, -140,  105,  42,  7]
%e A294034 [8][ 792, 1064, -1512, -840, -280, 168, 56, 8]
%p A294034 gf := exp(x*z)*z*((exp(z) + 1)/((exp(z) + 2*exp(-z/2)*cos(z*sqrt(3)/2))/3) - 1):
%p A294034 s := n -> n!*coeff(series(gf, z, 12), z, n):
%p A294034 C := n -> PolynomialTools:-CoefficientList(s(n), x):
%p A294034 ListTools:-FlattenOnce([seq(C(n), n=1..11)]);
%Y A294034 Cf. A003506, A294033.
%K A294034 sign,tabl
%O A294034 1,2
%A A294034 _Peter Luschny_, Oct 24 2017