A137437 Triangular sequence from expansion coefficients of asymptotic Hermite Polynomial from Roman: p(x,t)= (1 + t)^(-x)*Exp[x*(t - t^2/2)].

1, 0, 0, 0, -2, 0, 6, 0, -24, 0, 120, 40, 0, -720, -420, 0, 5040, 3948, 0, -40320, -38304, -2240, 0, 362880, 396576, 50400

Offset: 1

Author

Roger L. Bagula, Apr 21 2008

Keywords

• uned
• tabf
• sign

Comments

Absolute values of row sums give A038205.

Examples

			{1},
{0},
{0},
{0, -2},
{0, 6},
{0, -24},
{0, 120, 40},
{0, -720, -420},
{0, 5040, 3948},
{0, -40320, -38304, -2240},
{0, 362880, 396576, 50400}
		

References

• Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 130.

Crossrefs

Cf. A038205, A137286.

Programs

Mathematica

p[t_] := (1 + t)^(-x)*Exp[x*(t - t^2/2)];
Table[ ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}];
a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}];
Flatten[{{1}, {0}, {0}, {0, -2}, {0, 6}, {0, -24}, {0, 120, 40}, {0, -720, -420}, {0, 5040, 3948}, {0, -40320, -38304, -2240}, {0, 362880, 396576, 50400}}]

Formula

p(x,t)= (1 + t)^(-x)*Exp[x*(t - t^2/2)]=Sum[q(x,n)*t^n/n!,{n,0,Infinity}]; out(n,m)=n!*Coefficients(q(x,n)).