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

Author

Roger L. Bagula, May 03 2008

Keywords

• nonn
• uned
• tabl

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)).

Examples

			{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}
		

References

• A. Messiah, Quantum mechanics, vol. 2, p. 712, fig.XVIII.2, North Holland, 1969.

Crossrefs

Cf. A000522, A094587.

Programs

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]

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))