A158757 Expansion of e.g.f. exp(t*x)/(1 - x^2/t^2 - t^3* x^3).

1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 6, 0, 0, 0, 7, 24, 0, 0, 0, 12, 0, 0, 0, 25, 0, 0, 120, 0, 0, 0, 260, 0, 0, 0, 61, 720, 0, 0, 0, 360, 0, 0, 0, 1470, 0, 0, 0, 841, 0, 0, 5040, 0, 0, 0, 15960, 0, 0, 0, 5082, 0, 0, 0, 5251, 40320, 0, 0, 0, 20160, 0, 0, 0, 122640, 0, 0, 0, 134456, 0, 0, 0, 20497

Offset: 0

Author

Roger L. Bagula, Mar 25 2009

Keywords

• nonn
• tabf

Examples

			Irregular triangle begins as:
      1;
      0, 0,    1;
      2, 0,    0, 0,   1;
      0, 0,    6, 0,   0, 0,     7;
     24, 0,    0, 0,  12, 0,     0, 0,   25;
      0, 0,  120, 0,   0, 0,   260, 0,    0, 0,   61;
    720, 0,    0, 0, 360, 0,     0, 0, 1470, 0,    0, 0, 841;
      0, 0, 5040, 0,   0, 0, 15960, 0,    0, 0, 5082, 0,   0, 0, 5251;
		

References

• H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, page 221.

Links

• G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Crossrefs

Cf. A005212, A005359, A158706, A330044.

Programs

Mathematica

Table[CoefficientList[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/(1 - x^2/t^2 - t^3*x^3), {x, 0, 20}], n]], t], {n, 0, 10}]//Flatten

Sage

f(x,t) = exp(t*x)/(1 - x^2/t^2 - t^3*x^3)
def A158757(n,k): return ( factorial(n)*t^n*( f(x,t) ).series(x, 20).list()[n] ).series(t,2*n+1).list()[k]
flatten([[A158757(n,k) for k in (0..2*n)] for n in (0..10)]) # G. C. Greubel, Dec 05 2021

Formula

T(n, k) = coefficients of e.g.f.: exp(t*x)/(1 - x^2/t^2 - t^3* x^3).

From G. C. Greubel, Dec 05 2021: (Start)

T(n, 2*n) = A330044(n).

T(n, 0) = A005359(n).

T(n, 2) = A005212(n). (End)

Extensions

Edited by G. C. Greubel, Dec 01 2021