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Search: id:A138108
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| A138108 |
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A triangular sequence of coefficients based on the expansion of an Hamiltonian resolvent or Green's function: p(x,t)=Exp[x*t]/(x-t); where t is taken as the Hamiltonian variable and x as the complex variable. |
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1
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| 1, 1, 0, 1, 2, 0, 2, 0, 1, 6, 0, 6, 0, 3, 0, 1, 24, 0, 24, 0, 12, 0, 4, 0, 1, 120, 0, 120, 0, 60, 0, 20, 0, 5, 0, 1, 720, 0, 720, 0, 360, 0, 120, 0, 30, 0, 6, 0, 1, 5040, 0, 5040, 0, 2520, 0, 840, 0, 210, 0, 42, 0, 7, 0, 1, 40320, 0, 40320, 0, 20160, 0, 6720, 0, 1680, 0, 336, 0, 56, 0, 8
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Row sums are:
{1, 2, 5, 16, 65, 326, 1957, 13700, 109601, 986410, 9864101};
If you use a transform of;
x->Sqrt[y];
you get A094587.
The wave function form of the Green's function is:
G(x)*Phi[x,n]=Phi[x,n]/(x-E(n)).
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REFERENCES
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A. Messiah, Quantum mechanics, vol. 2, p. 712, fig.XVIII.2, North Holland, 1969.
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FORMULA
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p(x,t)=Exp[x*t]/(x-t)=sum(P(x,n)*t^n/n!,{n,0,Infinity}); Out_n,m=n!Coefficients(x^(n+1)*P(x,n))
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EXAMPLE
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{1},
{1, 0, 1},
{2, 0, 2, 0, 1},
{6, 0, 6, 0, 3, 0, 1},
{24, 0, 24, 0, 12, 0, 4, 0, 1},
{120, 0, 120, 0, 60, 0, 20, 0, 5, 0, 1},
{720, 0, 720, 0, 360, 0, 120, 0, 30, 0, 6, 0, 1},
{5040, 0, 5040, 0, 2520, 0, 840, 0, 210, 0, 42, 0, 7, 0, 1},
{40320, 0, 40320, 0, 20160, 0, 6720, 0, 1680, 0, 336, 0, 56, 0, 8, 0, 1}, {362880, 0, 362880, 0, 181440, 0, 60480, 0, 15120, 0, 3024, 0, 504, 0, 72, 0, 9, 0, 1},
{3628800, 0, 3628800, 0, 1814400, 0, 604800, 0, 151200, 0, 30240, 0, 5040, 0, 720, 0, 90, 0, 10, 0, 1}
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MATHEMATICA
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p[t_] = Exp[x*t]/(x - t); Table[ ExpandAll[x^(n + 1)*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[n!* CoefficientList[( x^(n + 1)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]), x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
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Cf. A000522, A094587.
Sequence in context: A022879 A064984 A038555 this_sequence A158777 A039970 A105118
Adjacent sequences: A138105 A138106 A138107 this_sequence A138109 A138110 A138111
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KEYWORD
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nonn,uned,tabl
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 03 2008
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