Hello.
I have a generating function G(t) = E^(-2*t^2)
How I can find using wolfram generate term of sequence which generates this function?
Welcome to the Wolfram Community,
Have you looked at the functions Series and InverseSeries and their help? There must be an example that you can use.
I’m not sure if your question is posed correctly; you want to generate terms that generates the function or generate terms that the funtion generates?
I have generating function G(t)
I want to know how looks a_n
for example
if i have G(t) = 1/(1-t)
a_n = 1 for any n
Table[SeriesCoefficient[1/(1 - t), {t, 0, n}], {n, 1, 10}]
You can query specific coefficients of the series using SeriesCoefficient. What have you tried yourself?
Table[SeriesCoefficient[e^(-2t^2), {t, 0, n}], {n, 0, 10}]
It gives me table of values for each n
but I want to get a formula for a_n depends on n.
Please have a look at GeneratingFunction, FindGeneratingFunction, FindSequenceFunction, SeriesCoefficient and Series there are a plethora of examples!
Thx for help.
I got it now! c:
Perhaps you can share your findings along with an example. Such as to make this discussion useful for future reference.
Hi Roman,
the sequence which generates the function is just the Taylor serie. The general formula for the coefficients are the well known terms \left.\frac{1}{n!}\frac{\partial}{\partial x} g(x)\right|_{x=0}. You can calculate a bunch of those terms and than use FindSequenceFunction to find in a somewhat empirical way a more specific formula:
g[t_] := Exp[-2*t^2]
gterms = Table[(D[g[t], {t, n}]/n!) /. t -> 0, {n, 0, 20}]
(* or likewise: *)
gterms = Table[SeriesCoefficient[g[t], {t, 0, n}], {n, 0, 20}]
(* find a specific formula for the terms: *)
FindSequenceFunction[gterms, n]
resulting in:
Isn’t Mathematica just great?!? Regards – Henrik
… How I can find using wolfram generate term of sequence
Hi. There is also CoefficientList … (which eliminates the table command) :>)
CoefficientList[Series[g[t],{t,0,20}],t]
{1,0,-2,0,2,0,-(4/3),0,2/3,... etc}