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%I #2 Mar 30 2012 17:34:26
%S 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,
%T 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,
%U 840,0,210,0,42,0,7,0,1,40320,0,40320,0,20160,0,6720,0,1680,0,336,0,56,0,8
%N 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.
%C Row sums are:
%C {1, 2, 5, 16, 65, 326, 1957, 13700, 109601, 986410, 9864101};
%C If you use a transform of;
%C x->Sqrt[y];
%C you get A094587.
%C The wave function form of the Green's function is:
%C G(x)*Phi[x,n]=Phi[x,n]/(x-E(n)).
%D A. Messiah, Quantum mechanics, vol. 2, p. 712, fig.XVIII.2, North Holland, 1969.
%F 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))
%e {1},
%e {1, 0, 1},
%e {2, 0, 2, 0, 1},
%e {6, 0, 6, 0, 3, 0, 1},
%e {24, 0, 24, 0, 12, 0, 4, 0, 1},
%e {120, 0, 120, 0, 60, 0, 20, 0, 5, 0, 1},
%e {720, 0, 720, 0, 360, 0, 120, 0, 30, 0, 6, 0, 1},
%e {5040, 0, 5040, 0, 2520, 0, 840, 0, 210, 0, 42, 0, 7, 0, 1},
%e {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},
%e {3628800, 0, 3628800, 0, 1814400, 0, 604800, 0, 151200, 0, 30240, 0, 5040, 0, 720, 0, 90, 0, 10, 0, 1}
%t 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]
%Y Cf. A000522, A094587.
%K nonn,uned,tabl
%O 1,5
%A _Roger L. Bagula_, May 03 2008
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