A138106 - OEIS

%I #9 Apr 02 2019 08:06:57

%S -1,0,-1,2,0,-1,-6,6,0,-1,14,-24,12,0,-1,-30,70,-60,20,0,-1,62,-180,

%T 210,-120,30,0,-1,-126,434,-630,490,-210,42,0,-1,254,-1008,1736,-1680,

%U 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

%N 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)).

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

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

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

%H G. C. Greubel, <a href="/A138106/b138106.txt">Rows n = 1..100 of triangle, flattened</a>

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

%e Triangle begins as:

%e     -1;

%e      0,    -1;

%e      2,     0,    -1;

%e     -6,     6,     0,     -1;

%e     14,   -24,    12,      0,    -1;

%e    -30,    70,   -60,     20,     0,    -1;

%e     62,  -180,   210,   -120,    30,     0,   -1;

%e   -126,   434,  -630,    490,  -210,    42,    0,   -1;

%e    254, -1008,  1736,  -1680,   980,  -336,   56,    0, -1;

%e   -510,  2286, -4536,   5208, -3780,  1764, -504,   72,  0, -1;

%e   1022, -5100, 11430, -15120, 13020, -7560, 2940, -720, 90,  0, -1;

%e   .....

%t p[t_] = Exp[x*t]*(Exp[ -2*t] - 2*Exp[ -t]);

%t Table[ ExpandAll[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}];

%t Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]//Flatten

%K tabl,sign

%O 1,4

%A _Roger L. Bagula_, May 03 2008

%E Edited by _G. C. Greubel_, Apr 01 2019