323557.lean`

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

7 declarations, 1 contain sorry, 0 declared as axioms

Pronic indices for odd coefficients of $\sum_{n \ge 0} x^n \frac{(1+x^n)^n}{(1+x^{n+1})^{n+1}}$

Coefficients of G.f. $\sum_{n\ge 0} x^n \cdot \frac{(1 + x^n)^n}{(1 + x^{n+1})^{n+1}}$. The $m$-th term $a(m)$ is the coefficient of $x^m$, which is explicitly given by the sum: $$ a(m) = \sum_{n=0}^m \sum_{k=0}^n \binom{n}{k} (-1)^j \binom{n+j}{j},$$ where $j = \frac{m - n(k+1)}{n+1}$, and the term is zero unless $j$ is a natural number.

References:

A323557
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA323557.a Coefficients of G.f line 46 source
Source documentation
Coefficients of G.f. $\sum_{n\ge 0} x^n \cdot \frac{(1 + x^n)^n}{(1 + x^{n+1})^{n+1}}$. The $m$-th term $a(m)$ is the coefficient of $x^m$, which is explicitly given by the sum: $$ a(m) = \sum_{n=0}^m \sum_{k=0}^n \binom{n}{k} (-1)^j \binom{n+j}{j},$$ where $j = \frac{m - n(k+1)}{n+1}$, and the term is zero unless $j$ is a natural number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, ite, LE.le, HSub.hSub, Dvd.dvd, HDiv.hDiv, Nat.choose, Even, Neg.neg
20 instances, coercions and auxiliary constants left out.
```
def a (m : ℕ) : ℤ :=
…
```
lemma OeisA323557.a_0 line 65 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA323557.a, OfNat.ofNat, Nat
Proof uses: Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA323557.a_1 line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA323557.a, OfNat.ofNat, Nat
Proof uses: Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA323557.a_2 line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA323557.a, OfNat.ofNat, Nat
Proof uses: Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA323557.a_3 line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA323557.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA323557.a_4 line 77 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA323557.a, OfNat.ofNat, Nat
Proof uses: Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA323557.odd_a_implies_pronic sorry line 92 source
depends on sorry what this means
Statement uses: Nat, Odd, Int, OeisA323557.a, Exists, Eq, HMul.hMul, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.