Symbolic computation of a series expansion | Cauchy's theorem

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Let us assume we do not know the series expansion at $0$ of $$S : z\mapsto\frac{1}{1-(z/2)^2}$$ (consider a more difficult rational fraction if you do not want to make this assumption).

A method to compute the coefficients (*) is to compute a complex integral on a circle: the $k$-th coefficient is given by $$ a(k)=\frac{1}{2i\pi}\int_{C(0,1)}\frac{S(z)}{z^{k+1}}dz=\int_0^1S\left(\exp\left(2i\pi t\right)\right)\exp\left(-2i\pi kt\right) dt. $$

But, even in this easy case, I am not able to help Sagemath to do the computation:

kk,k,t,z=var('kk,k,t,z')
assume(k,'integer')
assume(k>0)
S(z)=1/(1-(z/2)^2)
J(kk)=integrate(S(e^(2*I*pi*t))*e^(-2*pi*I*kk*t),t,0,1,hold=True)

Appealing J(k) returns a mistake and requires an assumption on $I$ and $\pi$ :

ValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(%i*%pi>0)', see `assume?` for more details)
Is %i*%pi an integer?

1. Any idea to perform the computation of this integral for any integer $k$ (I would like to have a symbolic answer in term of $k$, not a procedure that returns a numerical result for any numeric value of $k$)?
2. Is there another way to get the coefficients (again meaning a general formula depending on k, like 1/k! for exp).

(*) Not the only one, one also could you a general result on the form of a sequence whose generating function is a rational fraction.

NB. A similar question was asked a few years ago, without positive answer. But situation might have evolved.

The full text Sage message :

---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/interfaces/maxima_lib.py:817, in MaximaLib.sr_integral(self, *args)
    816 try:
--> 817     return max_to_sr(maxima_eval(([max_integrate],
    818                                   [sr_to_max(SR(a)) for a in args])))
    819 except RuntimeError as error:

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/libs/ecl.pyx:838, in sage.libs.ecl.EclObject.__call__ (build/cythonized/sage/libs/ecl.c:11700)()
    837 lispargs = EclObject(list(args))
--> 838 return ecl_wrap(ecl_safe_apply(self.obj, (<EclObject>lispargs).obj))
    839 

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/libs/ecl.pyx:358, in sage.libs.ecl.ecl_safe_apply (build/cythonized/sage/libs/ecl.c:9207)()
    357     else:
--> 358         raise RuntimeError("ECL says: {}".format(message))
    359 else:

RuntimeError: ECL says: Maxima asks: Is %i*%pi an integer?

During handling of the above exception, another exception occurred:

ValueError                                Traceback (most recent call last)
Cell In[4], line 1
----> 1 J(k)

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/expression.pyx:6187, in sage.symbolic.expression.Expression.__call__ (build/cythonized/sage/symbolic/expression.cpp:78521)()
   6185         z^2 + x^y
   6186     """
-> 6187     return self._parent._call_element_(self, *args, **kwds)
   6188 
   6189 def variables(self):

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/callable.py:422, in CallableSymbolicExpressionRing_class._call_element_(self, _the_element, *args, **kwds)
    420 d = dict(zip([repr(_) for _ in self.arguments()], args))
    421 d.update(kwds)
--> 422 return SR(_the_element.substitute(**d))

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/structure/element.pyx:849, in sage.structure.element.Element.substitute (build/cythonized/sage/structure/element.c:15440)()
    847     5
    848 """
--> 849 return self.subs(*args, **kwds)
    850 
    851 

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/expression.pyx:5899, in sage.symbolic.expression.Expression.subs (build/cythonized/sage/symbolic/expression.cpp:76733)()
   5897     smap.insert(make_pair((<Expression>self.coerce_in(k))._gobj,
   5898                           (<Expression>self.coerce_in(v))._gobj))
-> 5899 res = self._gobj.subs_map(smap, 0)
   5900 return new_Expression_from_GEx(self._parent, res)
   5901 

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/function.pyx:1137, in sage.symbolic.function.BuiltinFunction._evalf_or_eval_ (build/cythonized/sage/symbolic/function.c:16410)()
   1135 res = self._evalf_try_(*args)
   1136 if res is None:
-> 1137     return self._eval0_(*args)
   1138 else:
   1139     return res

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/integration/integral.py:254, in DefiniteIntegral._eval_(self, f, x, a, b)
    252 for integrator in self.integrators:
    253     try:
--> 254         A = integrator(*args)
    255     except (NotImplementedError, TypeError,
    256             AttributeError, RuntimeError):
    257         pass

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/integration/external.py:46, in maxima_integrator(expression, v, a, b)
     44     result = maxima.sr_integral(expression, v)
     45 else:
---> 46     result = maxima.sr_integral(expression, v, a, b)
     47 return result._sage_()

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/interfaces/maxima_lib.py:827, in MaximaLib.sr_integral(self, *args)
    825     raise ValueError("Integral is divergent.")
    826 elif "Is" in s:  # Maxima asked for a condition
--> 827     self._missing_assumption(s)
    828 else:
    829     raise

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/interfaces/maxima_lib.py:1074, in MaximaLib._missing_assumption(self, errstr)
   1071 outstr = "Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume("\
   1072     + errstr[jj + 1:k] + ">0)', see `assume?` for more details)\n" + errstr
   1073 outstr = outstr.replace('_SAGE_VAR_', '')
-> 1074 raise ValueError(outstr)

ValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(%i*%pi>0)', see `assume?` for more details)
Is %i*%pi an integer?