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.

1, -1, 2, -1, -2, 2, 1, -6, -6, 4, 10, 4, -12, -8, 4, 26, 100, 20, -40, -20, 8, -154, 156, 300, 40, -60, -24, 8, -1646, -2156, 1092, 1400, 140, -168, -56, 16, 1000, -13168, -8624, 2912, 2800, 224, -224, -64, 16, 92744, 18000, -118512, -51744, 13104, 10080, 672, -576, -144, 32

Offset: 0

Author

Roger L. Bagula, Dec 15 2009

Keywords

• sign
• tabl

Comments

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+...

Row n contains the coefficients of [x^k] of the polynomial in front of t^n, multiplied by 4*floor((n+1)/2)*n!.

Row sums are: 1, 1, -1, -7, -2, 94, 266, -1378, -15128, -36344, 839144,...

Examples

			The triangle starts in row n=0, columns 0<=k <=n as
      1;
     -1,      2;
     -1,     -2,       2;
      1,     -6,      -6,      4;
     10,      4,     -12,     -8,     4;
     26,    100,      20,    -40,   -20,     8;
   -154,    156,     300,     40,   -60,   -24,    8;
  -1646,  -2156,    1092,   1400,   140,  -168,  -56,   16;
   1000, -13168,   -8624,   2912,  2800,   224, -224,  -64,   16;
  92744,  18000, -118512, -51744, 13104, 10080,  672, -576, -144, 32;
  ...
		

Crossrefs

Cf. A000364, A060083, A171684.

Programs

Mathematica

Clear[p, g, m, a];
m = 1;
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}]
a = Table[CoefficientList[FullSimplify[ExpandAll[2^Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}]
Flatten[a]

Extensions

Number of variables in use reduced from 4 to 2, keyword:tabl added - The Assoc. Eds. of the OEIS, Oct 20 2010