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.

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.

Links

Table of n, a(n) for n=1..79.

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 A373917

Adjacent sequences: A138105 A138106 A138107 * A138109 A138110 A138111

Keyword

nonn,uned,tabl

Author

Roger L. Bagula, May 03 2008

Status

approved