A138106 A triangular sequence of coefficients based on the expansion of a Morse potential type function: p(x,t) = exp(x*t)*(exp(-2*t) - 2*exp(-t)).

-1, 0, -1, 2, 0, -1, -6, 6, 0, -1, 14, -24, 12, 0, -1, -30, 70, -60, 20, 0, -1, 62, -180, 210, -120, 30, 0, -1, -126, 434, -630, 490, -210, 42, 0, -1, 254, -1008, 1736, -1680, 980, -336, 56, 0, -1, -510, 2286, -4536, 5208, -3780, 1764, -504, 72, 0, -1, 1022, -5100, 11430, -15120, 13020, -7560, 2940, -720, 90, 0, -1

Offset: 1

Author

Roger L. Bagula, May 03 2008

Keywords

• tabl
• sign

Comments

Row sums are: {-1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1,...}.

The Morse potential is identified with simple intermolecular energy to distance relationships.

Examples

			Triangle begins as:
    -1;
     0,    -1;
     2,     0,    -1;
    -6,     6,     0,     -1;
    14,   -24,    12,      0,    -1;
   -30,    70,   -60,     20,     0,    -1;
    62,  -180,   210,   -120,    30,     0,   -1;
  -126,   434,  -630,    490,  -210,    42,    0,   -1;
   254, -1008,  1736,  -1680,   980,  -336,   56,    0, -1;
  -510,  2286, -4536,   5208, -3780,  1764, -504,   72,  0, -1;
  1022, -5100, 11430, -15120, 13020, -7560, 2940, -720, 90,  0, -1;
  .....
		

References

• A. Messiah, Quantum mechanics, vol. 2, p. 795, fig.XVIII.2, North Holland, 1969.

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Programs

Mathematica

p[t_] = Exp[x*t]*(Exp[ -2*t] - 2*Exp[ -t]);
Table[ ExpandAll[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}];
Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]//Flatten

Formula

p(x,t) = exp(x*t)*(exp(-2*t) - 2*exp(-t)) = Sum_{n>=0} P(x,n)*t^n/n!.

Extensions

Edited by G. C. Greubel, Apr 01 2019