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%I
%S 1,-1,2,-3,-2,2,-11,-18,-6,4,30,-44,-36,-8,4,866,300,-220,-120,-20,8,
%T 3858,5196,900,-440,-180,-24,8,-23654,54012,36372,4200,-1540,-504,-56,
%U 16,-722760,-189232,216048,96992,8400,-2464,-672,-64,16,-10842136
%N Triangle T(n,k) which contains 8*n!*2^floor((n+1)/2) times the coefficient [t^n x^k] exp(t*x)/(7 + exp(4*t)) in row n, column k.
%C The bivariate taylor expansion of exp(t*x)/(7+exp(4*t)) is 1/8 + (x/8-1/16)*t +(-3/32+x^2/16 -x/16)*t^2+...
%C Row n contains the coefficients of the polynomial in front of t^n, multiplied by 8*floor[(n+1)/2]*n!.
%C Row sums are: 1, 1, -3, -31, -54, 814, 9318, 68846, -593736, -23801144, -146144808, ....
%e The triangle starts in row n=0 with columns 0<=k <=n as
%e 1;
%e -1, 2;
%e -3, -2, 2;
%e -11, -18, -6, 4;
%e 30, -44, -36, -8, 4;
%e 866, 300, -220, -120, -20, 8;
%e 3858, 5196, 900, -440, -180, -24, 8;
%e -23654, 54012, 36372, 4200, -1540, -504, -56, 16;
%e -722760, -189232, 216048, 96992, 8400, -2464, -672, -64, 16;
%e -10842136, -13009680, -1703088, 1296288, 436464, 30240, -7392, -1728, -144, 32;
%e 28850712, -108421360, -65048400, -5676960, 3240720, 872928, 50400, -10560, -2160, -160, 32;
%t Clear[p, g, m, a];
%t m = 2;
%t p[t_] = 2^(m + 1)*Exp[t*x]/(-1 + 2^(m + 1) + Exp[2^m*t])
%t Table[ FullSimplify[ExpandAll[2^ Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], {n, 0, 10}]
%t a = Table[CoefficientList[FullSimplify[ExpandAll[2^Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}]
%t Flatten[a]
%Y Cf. A000364, A171683.
%K sign,tabl
%O 0,3
%A _Roger L. Bagula_, Dec 15 2009
%E Number of variables in use reduced from 4 to 2, keyword:tabl added - The Assoc. Eds. of the OEIS, Oct 20 2010
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