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 A138108 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. 0
 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; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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)). REFERENCES A. Messiah, Quantum mechanics, vol. 2, p. 712, fig.XVIII.2, North Holland, 1969. LINKS FORMULA 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)) EXAMPLE {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} MATHEMATICA 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] CROSSREFS Cf. A000522, A094587. Sequence in context: A307197 A328949 A038555 * A158777 A039970 A179212 Adjacent sequences:  A138105 A138106 A138107 * A138109 A138110 A138111 KEYWORD nonn,uned,tabl AUTHOR Roger L. Bagula, May 03 2008 STATUS approved

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Last modified September 24 04:42 EDT 2020. Contains 337317 sequences. (Running on oeis4.)