Lazy power series: fix bad handling of base ring and categorification

[hivert]

Currently

sage: R = LazyPowerSeriesRing(QQ)
sage: type(R.gen().coefficient(0))
<type 'sage.rings.rational.Rational'>
sage: type(R.gen().coefficient(1))
<type 'int'>
sage: type(R.gen().coefficient(2))
<type 'int'>

it should be always Rational

I also fixed the following bug:

sage: from sage.combinat.species.series import LazyPowerSeriesRing
sage: L = LazyPowerSeriesRing(QQ)
sage: g = L.gen(); z = L.zero()
sage: s = z+g; s
Uninitialized lazy power series
sage: s.coefficients(10)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
sage: s
Uninitialized lazy power series

It should be:

sage: s.coefficients(10)
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
sage: s
x + O(x^10)

Depends on #14685

CC: @sagetrac-sage-combinat @simon-king-jena @fchapoton @darijgr

Component: combinatorics

Keywords: LazyPowerSeries, Species, days49

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/13433

[hivert]

comment:1

The patch is not yet ready for review !

[hivert]

Description changed:

--- 
+++ 
@@ -11,3 +11,25 @@
 ```
 it should be always `Rational`
 
+I also fixed the following bug:
+
+```
+sage: from sage.combinat.species.series import LazyPowerSeriesRing
+sage: L = LazyPowerSeriesRing(QQ)
+sage: g = L.gen(); z = L.zero()
+sage: s = z+g; s
+Uninitialized lazy power series
+sage: s.coefficients(10)
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+sage: s
+Uninitialized lazy power series
+```
+It should be:
+
+```
+sage: s.coefficients(10)
+[0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
+sage: s
+x + O(x^10)
+```
+

[hivert]

Changed keywords from LazyPowerSeries to LazyPowerSeries, Species

[nthiery]

comment:5

Salut Florent,

There are some failing tests:

sage -t sage/combinat/tutorial.py  # 5 doctests failed
sage -t sage/tests/french_book/polynomes.py  # 7 doctests failed

Could you fix those or move the patch down the queue?

Thanks,
Nicolas

[hivert]

comment:6

Fixed !

Florent

[hivert]

comment:8

Fixed a multiline doctest to silence the patchbot. Ready for review.

Florent

[hivert]

Changed keywords from LazyPowerSeries, Species to LazyPowerSeries, Species, days49