A176295 Triangle read by rows, based on the two-variable g.f. exp(x*t)*(x*(1 - 2*exp(x)) - 2*exp(x))/(1 - exp(t)) (the second of two parts).

-4, 4, 8, 2, -10, 0, 12, 0, 16, -32, -16, 32, -4, -4, 120, -120, -120, 120, 0, -96, -96, 960, -480, -864, 576, 80, 80, -1680, -1680, 8400, -1680, -6720, 3360, 0, 3840, 3840, -26880, -26880, 80640, 0, -57600, 23040, -6048, -6048, 120960, 120960, -423360, -423360, 846720, 120960, -544320, 181440

Offset: 0

Author

Roger L. Bagula, Dec 07 2010

Keywords

• sign
• tabf

Comments

A factor of 2*n!*(n+2)! was used to make the expansion coefficients all integers. This part is the b(i) part of the Sum_{j=0..n} (a(i) + b(i)*Exp(x) )*x^i, expansion.

Row sums are {8, 4, 0, -8, 0, 160, 0, -12096, 0, 2419200, 0,....}.

Examples

			Triangle begins as:
  -4,    4,     8;
   2,  -10,     0,     12;
   0,   16,   -32,    -16,     32;
  -4,   -4,   120,   -120,   -120,   120;
   0,  -96,   -96,    960,   -480,  -864,   576;
  80,   80, -1680,  -1680,   8400, -1680, -6720,   3360;
   0, 3840,  3840, -26880, -26880, 80640,     0, -57600, 23040;
		

References

• Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965, pp 296-298

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A048998, A138133 (the first part of the expansion).

Programs

Mathematica

p[t_]:= Exp[x*t]*(x*(1 -2*Exp[x]) -2*Exp[x])/(1-Exp[t]); Table[Im[ CoefficientList[2*n!*(n+2)!*SeriesCoefficient[Series[p[t], {t,0,30}]/.Exp[x] -> I, n], x]], {n,0,12}]//Flatten

Extensions

Edited by N. J. A. Sloane, Jan 01 2011