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A158757
Expansion of e.g.f. exp(t*x)/(1 - x^2/t^2 - t^3* x^3).
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
0,5
REFERENCES
H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, page 221.
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)
EXAMPLE
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;
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
PROG
(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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula, Mar 25 2009
EXTENSIONS
Edited by G. C. Greubel, Dec 01 2021
STATUS
approved