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A138160
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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]).
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0
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1, -1, 1, -1, 4, -4, 3, -2, 1, -24, 36, -27, 13, -6, 3, -1, 182, -354, 330, -198, 85, -28, 10, -4, 1, -1730, 4090, -4480, 3120, -1595, 631, -195, 50, -15, 5, -1, 19802, -55270, 70430, -55730, 31630, -14018, 5101, -1536, 375, -80, 21, -6, 1, -264334, 850990, -1246504, 1121960, -711480, 345268, -135639
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OFFSET
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1,5
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COMMENTS
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The row sums are:A133942; (-1)^n*n!;
{1, -1, 2, -6, 24, -120, 720, -5040, 40320, -362880, 3628800};
Plotting the relative Kinetic curves gives even peaks and odd Serpentine curves:
s = Table[Plot[g[[n]]/(1 - x)^(n + 1), {x, 0, 10}], {n, 1, Length[g]}].
It appears this method solves the spectrum of relative excited states for a Morse potential Hamiltonian system.
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REFERENCES
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A. Messiah, Quantum mechanics, vol. 2, page 712, 795-800, North Holland, 1969.
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LINKS
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FORMULA
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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)).
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EXAMPLE
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{1},
{-1, 1, -1},
{4, -4, 3, -2, 1},
{-24,36, -27, 13, -6, 3, -1},
{182, -354, 330, -198, 85, -28, 10, -4, 1},
{-1730, 4090, -4480, 3120, -1595, 631, -195, 50, -15, 5, -1},
{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},
{4031750, -14703010, 24411590, -24803170, 17590846, -9435594,4041366, -1436802, 437613, -117412, 28112, -5852, 1022, -168, 36, -8,1},
{-69183618,281868618, -524145948, 595915740, -470444436, 278589780, -130600260, 50396868, -16475823, 4667551, -1170885, 265812, -54894, 9954, -1530,228, -45, 9, -1},
{1319095202, -5938151118, 12241920582, -15446908458, 13503504972, -8806408212, 4513236588, -1890091092, 666457818, -202452462, 53920021, -12778960, 2741385, -542340, 98850, -15912, 2190, -300, 55, -10, 1}
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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