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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
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.
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
Sequence in context: A307197 A328949 A038555 * A158777 A393358 A039970
KEYWORD
nonn,uned,tabl
AUTHOR
Roger L. Bagula, May 03 2008
STATUS
approved