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A171683 Triangle T(n,k) which contains 4*n!*2^floor((n+1)/2) times the coefficient [t^n x^k] exp(t*x)/(3 + exp(2*t)) in row n, column k.

%I A171683 #15 Sep 10 2024 00:21:59
%S A171683 1,-1,2,-1,-2,2,1,-6,-6,4,10,4,-12,-8,4,26,100,20,-40,-20,8,-154,156,
%T A171683 300,40,-60,-24,8,-1646,-2156,1092,1400,140,-168,-56,16,1000,-13168,
%U A171683 -8624,2912,2800,224,-224,-64,16,92744,18000,-118512,-51744,13104,10080,672,-576,-144,32
%N A171683 Triangle T(n,k) which contains 4*n!*2^floor((n+1)/2) times the coefficient [t^n x^k] exp(t*x)/(3 + exp(2*t)) in row n, column k.
%C A171683 The bivariate Taylor expansion of exp(t*x)/(3+exp(2*t)) is 1/4 + (x/4-1/8)*t +(-1/16+x^2/8-x/8)*t^2+...
%C A171683 Row n contains the coefficients of [x^k] of the polynomial in front of t^n, multiplied by 4*floor((n+1)/2)*n!.
%C A171683 Row sums are: 1, 1, -1, -7, -2, 94, 266, -1378, -15128, -36344, 839144,...
%e A171683 The triangle starts in row n=0, columns 0<=k <=n as
%e A171683       1;
%e A171683      -1,      2;
%e A171683      -1,     -2,       2;
%e A171683       1,     -6,      -6,      4;
%e A171683      10,      4,     -12,     -8,     4;
%e A171683      26,    100,      20,    -40,   -20,     8;
%e A171683    -154,    156,     300,     40,   -60,   -24,    8;
%e A171683   -1646,  -2156,    1092,   1400,   140,  -168,  -56,   16;
%e A171683    1000, -13168,   -8624,   2912,  2800,   224, -224,  -64,   16;
%e A171683   92744,  18000, -118512, -51744, 13104, 10080,  672, -576, -144, 32;
%e A171683   ...
%t A171683 Clear[p, g, m, a];
%t A171683 m = 1;
%t A171683 p[t_] = 2^(m + 1)*Exp[t*x]/(-1 + 2^(m + 1) + Exp[2^m*t]) Table[ FullSimplify[ExpandAll[2^ Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], {n, 0, 10}]
%t A171683 a = Table[CoefficientList[FullSimplify[ExpandAll[2^Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}]
%t A171683 Flatten[a]
%Y A171683 Cf. A000364, A060083, A171684.
%K A171683 sign,tabl
%O A171683 0,3
%A A171683 _Roger L. Bagula_, Dec 15 2009
%E A171683 Number of variables in use reduced from 4 to 2, keyword:tabl added - The Assoc. Eds. of the OEIS, Oct 20 2010