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A158785
Expansion of e.g.f.: exp(t*x)/(1 - x^2/t - t^3*x^3).
1
1, 0, 1, 2, 0, 0, 1, 0, 6, 0, 0, 7, 24, 0, 0, 12, 0, 0, 25, 0, 120, 0, 0, 260, 0, 0, 61, 720, 0, 0, 360, 0, 0, 1470, 0, 0, 841, 0, 5040, 0, 0, 15960, 0, 0, 5082, 0, 0, 5251, 40320, 0, 0, 20160, 0, 0, 122640, 0, 0, 134456, 0, 0, 20497, 0, 362880, 0, 0, 1512000
OFFSET
0,4
FORMULA
T(n, k) = coefficients of e.g.f.: t^floor(n/2)*exp(t*x)/(1 - x^2/t - t^3*x^3).
From G. C. Greubel, Dec 05 2021: (Start)
T(n, floor(n/2) + n) = A330044(n).
T(n, 0) = A005359(n).
T(n, 1) = A005212(n). (End)
EXAMPLE
Irregular triangle begins as:
1;
0, 1;
2, 0, 0, 1;
0, 6, 0, 0, 7;
24, 0, 0, 12, 0, 0, 25;
0, 120, 0, 0, 260, 0, 0, 61;
720, 0, 0, 360, 0, 0, 1470, 0, 0, 841;
0, 5040, 0, 0, 15960, 0, 0, 5082, 0, 0, 5251;
40320, 0, 0, 20160, 0, 0, 122640, 0, 0, 134456, 0, 0, 20497;
MATHEMATICA
Table[CoefficientList[Expand[t^Floor[n/2]*n!*SeriesCoefficient[Series[Exp[t*x]/(1 - x^2/t - 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 - t^3*x^3)
def A158785(n, k): return ( factorial(n)*t^(n//2)*( f(x, t) ).series(x, 20).list()[n] ).series(t, 2*n+1).list()[k]
flatten([[A158785(n, k) for k in (0..n+(n//2))] for n in (0..10)]) # G. C. Greubel, Dec 05 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Mar 26 2009
EXTENSIONS
Edited by G. C. Greubel, Dec 05 2021
STATUS
approved