A138160 - OEIS

%I #6 Sep 07 2013 21:14:41

%S 1,-1,1,-1,4,-4,3,-2,1,-24,36,-27,13,-6,3,-1,182,-354,330,-198,85,-28,

%T 10,-4,1,-1730,4090,-4480,3120,-1595,631,-195,50,-15,5,-1,19802,

%U -55270,70430,-55730,31630,-14018,5101,-1536,375,-80,21,-6,1,-264334,850990,-1246504,1121960,-711480,345268,-135639

%N A triangular sequence of coefficients of the expansion of a Green's function for the radial Morse potential with x being the kinetic energy and t being the radius: Hamiltonian; H=K+V=x+Exp[-2*t]-2*Exp[t];G*Exp[t*x)=Exp[x*t]/(t-H); p(t,x)=Exp[t*x]/(t-x-Exp[-2*t] + 2*Exp[-t]).

%C The row sums are:A133942; (-1)^n*n!;

%C {1, -1, 2, -6, 24, -120, 720, -5040, 40320, -362880, 3628800};

%C Plotting the relative Kinetic curves gives even peaks and odd Serpentine curves:

%C s = Table[Plot[g[[n]]/(1 - x)^(n + 1), {x, 0, 10}], {n, 1, Length[g]}].

%C It appears this method solves the spectrum of relative excited states for a Morse potential Hamiltonian system.

%D A. Messiah, Quantum mechanics, vol. 2, page 712, 795-800, North Holland, 1969.

%F p(t,x)=Exp[t*x]/(t - x - Exp[ -2*t] + 2*Exp[ -t])=sum(P(x,n)*t^n/n!,m{n,0,Infinity}); out_n,m=Coefficients(n!*(1-x)^(n+1)*P(x,n)).

%e {1},

%e {-1, 1, -1},

%e {4, -4, 3, -2, 1},

%e {-24,36, -27, 13, -6, 3, -1},

%e {182, -354, 330, -198, 85, -28, 10, -4, 1},

%e {-1730, 4090, -4480, 3120, -1595, 631, -195, 50, -15, 5, -1},

%e {19802, -55270, 70430, -55730, 31630, -14018, 5101, -1536, 375, -80, 21, -6,1}, {-264334, 850990, -1246504, 1121960, -711480,345268, -135639, 44997, -12922, 3171, -644, 119, -28, 7, -1},

%e {4031750, -14703010, 24411590, -24803170, 17590846, -9435594,4041366, -1436802, 437613, -117412, 28112, -5852, 1022, -168, 36, -8,1},

%e {-69183618,281868618, -524145948, 595915740, -470444436, 278589780, -130600260, 50396868, -16475823, 4667551, -1170885, 265812, -54894, 9954, -1530,228, -45, 9, -1},

%e {1319095202, -5938151118, 12241920582, -15446908458, 13503504972, -8806408212, 4513236588, -1890091092, 666457818, -202452462, 53920021, -12778960, 2741385, -542340, 98850, -15912, 2190, -300, 55, -10, 1}

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

%K uned,tabl,sign

%O 1,5

%A _Roger L. Bagula_, May 04 2008