2454.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

Central factorial numbers: $((2n)!!)^2$

Central factorial numbers: $a(n) = 4^n (n!)^2 = ((2n)!!)^2$.

References:

Imports (1)

FormalConjecturesUtil

def OeisA2454.a Central factorial numbers: $a(n) = 4^n (n!)^2$ line 35 source
Source documentation
Central factorial numbers: $a(n) = 4^n (n!)^2$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: HMul.hMul, HPow.hPow, OfNat.ofNat, Nat.factorial
7 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA2454.a_0 Value of the sequence a at 0 line 40 source
Source documentation
Value of the sequence a at 0.

standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA2454.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 1 := by rfl
…
```
Mentions: OeisA2454.a
theorem OeisA2454.a_1 Value of the sequence a at 1 line 44 source
Source documentation
Value of the sequence a at 1.

standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA2454.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 4 := by rfl
…
```
Mentions: OeisA2454.a
theorem OeisA2454.a_2 Value of the sequence a at 2 line 48 source
Source documentation
Value of the sequence a at 2.

standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA2454.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 64 := by rfl
…
```
Mentions: OeisA2454.a
theorem OeisA2454.a_3 Value of the sequence a at 3 line 52 source
Source documentation
Value of the sequence a at 3.

standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA2454.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 2304 := by rfl
…
```
Mentions: OeisA2454.a
theorem OeisA2454.a_4 Value of the sequence a at 4 line 56 source
Source documentation
Value of the sequence a at 4.

standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA2454.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 147456 := by rfl
…
```
Mentions: OeisA2454.a
theorem OeisA2454.conjecture Let $\zeta$ be a primitive $(2n+1)$-th root of unity sorry line 69 source
Source documentation
Let $\zeta$ be a primitive $(2n+1)$-th root of unity. Then the permanent of the $2n \times 2n$ matrix $[m(j,k)]_{j,k=1..2n}$ is $a(n)/(2n+1) = ((2n)!!)^2/(2n+1)$, where $m(j,k)$ is $1$ or $(1+\zeta^{j-k})/(1-\zeta^{j-k})$ according as $j = k$ or not.
- Zhi-Wei Sun, Jun 26 2022
Proved by [SSX22], Theorem 1.3(ii): for odd $m > 1$ and $\zeta$ a primitive $m$-th root of unity, the permanent of $[c_{j,k}]_{1 \le j,k \le m-1}$ is $((m-1)!!)^2/m$. Take $m = 2n+1$; translating both matrix indices by one leaves $j - k$ unchanged. For $n = 0$ the matrix is empty and both sides are $1$.
depends on sorry what this means
Statement uses: Nat, HMul.hMul, OfNat.ofNat, HAdd.hAdd, Complex, IsPrimitiveRoot, Eq, Matrix.permanent, Fin, ite, Int, HSub.hSub, HDiv.hDiv, HPow.hPow, OeisA2454.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) :
    let N : ℕ := 2 * n
…
```