Unable to extract a particular coefficient of a multivariate power series

[maxale]

Steps To Reproduce

We have

sage: R.<x,y> = QQ[]
sage: ((x+y)^3)[2,1]
3

but

sage: R.<x,y> = QQ[[]]
sage: ((x+y)^3)[2,1]
[...]
TypeError: unsupported operand parent(s) for <=: 'The Infinity Ring' and '<class 'tuple'>'

Expected Behavior

There should be a simple interface (including indexing via []) for accessing a particular coefficient of a power series, similarly to that of multivariate polynomials.

Actual Behavior

Functionality for accessing individual coefficients is missing.

Additional Information

No response

Environment

• OS: Ubuntu 24.04.1 LTS
• Sage Version: 10.6.beta1

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide

[DaveWitteMorris]

As far as I know, there is nothing in the documentation that says this should work, so it seems to be an enhancement request, not a bug.

[maxale]

Well, it also works for univariate series:

sage: R.<x> = QQ[[]]
sage: ((x+1)^3)[2]
3

So, the absence of this core functionality for multivariate series is a clear deficiency, if not a bug.

[mantepse]

Let me remark that, currently, __getitem__ extracts the piece of given degree for you:

sage: R.<x,y> = QQ[[]]
sage: (1/(1-x+y)^3)[4]
15*x^4 - 60*x^3*y + 90*x^2*y^2 - 60*x*y^3 + 15*y^4

This also works, and more generally, for lazy series:

sage: s = SymmetricFunctions(QQ).s()
sage: L = LazySymmetricFunctions(s)
sage: f = 1/(1-L(s[1]))
sage: f[4]
s[1, 1, 1, 1] + 3*s[2, 1, 1] + 2*s[2, 2] + 3*s[3, 1] + s[4]

We can now extract a single coefficient - note that providing an index of the wrong type silently returns 0.

sage: f[4][2,1,1]
0
sage: f[4][Partition([2,1,1])]
3

And, although, slightly abusing terminology,

sage: D = LazyDirichletSeriesRing(QQ, "s")
sage: D(lambda n: n)
1 + 2/2^s + 3/3^s + 4/4^s + 5/5^s + 6/6^s + 7/7^s + O(1/(8^s))
sage: f = D(lambda n: n)
sage: f[4]
4

[maxale]

Let me remark that, currently, __getitem__ extracts the piece of given degree for you:

Perhaps, for the uniformity of interface __getitem__ should support both an integer argument (the current behavior) and a tuple/list (of length equal the number of variables) argument, which will also eliminate the unnecessary step of extracting a homogeneous component when only a particular coefficient is needed.

[DaveWitteMorris]

The enhancement seems like a good idea. In any case, a way to obtain the coefficients (and the homogeneous components) should be documented, but doesn't seem to be.