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.

1, -1, 2, -3, -2, 2, -11, -18, -6, 4, 30, -44, -36, -8, 4, 866, 300, -220, -120, -20, 8, 3858, 5196, 900, -440, -180, -24, 8, -23654, 54012, 36372, 4200, -1540, -504, -56, 16, -722760, -189232, 216048, 96992, 8400, -2464, -672, -64, 16, -10842136, -13009680, -1703088, 1296288, 436464, 30240, -7392, -1728, -144, 32

Offset: 0

Author

Roger L. Bagula, Dec 15 2009

Keywords

• sign
• tabl

Comments

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+...

Row n contains the coefficients of the polynomial in front of t^n, multiplied by 8*floor((n+1)/2)*n!.

Row sums are: 1, 1, -3, -31, -54, 814, 9318, 68846, -593736, -23801144, -146144808, ....

Examples

			The triangle starts in row n=0 with columns 0<=k <=n as
        1;
       -1,       2;
       -3,      -2,      2;
      -11,     -18,     -6,     4;
       30,     -44,    -36,    -8,     4;
      866,     300,   -220,  -120,   -20,     8;
     3858,    5196,    900,  -440,  -180,   -24,    8;
   -23654,   54012,  36372,  4200, -1540,  -504,  -56,  16;
  -722760, -189232, 216048, 96992,  8400, -2464, -672, -64, 16;
  ...
		

Crossrefs

Cf. A000364, A171683.

Programs

Mathematica

Clear[p, g, m, a];
m = 2;
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