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A171684 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.

%I A171684 #15 Sep 09 2024 15:04:11
%S A171684 1,-1,2,-3,-2,2,-11,-18,-6,4,30,-44,-36,-8,4,866,300,-220,-120,-20,8,
%T A171684 3858,5196,900,-440,-180,-24,8,-23654,54012,36372,4200,-1540,-504,-56,
%U A171684 16,-722760,-189232,216048,96992,8400,-2464,-672,-64,16,-10842136,-13009680,-1703088,1296288,436464,30240,-7392,-1728,-144,32
%N A171684 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 A171684 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 A171684 Row n contains the coefficients of the polynomial in front of t^n, multiplied by 8*floor((n+1)/2)*n!.
%C A171684 Row sums are: 1, 1, -3, -31, -54, 814, 9318, 68846, -593736, -23801144, -146144808, ....
%e A171684 The triangle starts in row n=0 with columns 0<=k <=n as
%e A171684         1;
%e A171684        -1,       2;
%e A171684        -3,      -2,      2;
%e A171684       -11,     -18,     -6,     4;
%e A171684        30,     -44,    -36,    -8,     4;
%e A171684       866,     300,   -220,  -120,   -20,     8;
%e A171684      3858,    5196,    900,  -440,  -180,   -24,    8;
%e A171684    -23654,   54012,  36372,  4200, -1540,  -504,  -56,  16;
%e A171684   -722760, -189232, 216048, 96992,  8400, -2464, -672, -64, 16;
%e A171684   ...
%t A171684 Clear[p, g, m, a];
%t A171684 m = 2;
%t A171684 p[t_] = 2^(m + 1)*Exp[t*x]/(-1 + 2^(m + 1) + Exp[2^m*t])
%t A171684 Table[ FullSimplify[ExpandAll[2^ Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], {n, 0, 10}]
%t A171684 a = Table[CoefficientList[FullSimplify[ExpandAll[2^Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}]
%t A171684 Flatten[a]
%Y A171684 Cf. A000364, A171683.
%K A171684 sign,tabl
%O A171684 0,3
%A A171684 _Roger L. Bagula_, Dec 15 2009
%E A171684 Number of variables in use reduced from 4 to 2, keyword:tabl added - The Assoc. Eds. of the OEIS, Oct 20 2010