115257.lean`

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L4DEL: Lean4 Domain Explanation Language

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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

8 declarations, 1 contain sorry, 0 declared as axioms

Partial sums of $\binom{2n}{n}^2$

$$a(n) = \sum_{k=0}^n \binom{2k}{k}^2$$

References:

A115257

Imports (1)

FormalConjecturesUtil

def OeisA115257.a The primary defining sequence a line 39 source
Source documentation
The primary defining sequence a. Partial sums of $\binom{2n}{n}^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HPow.hPow, Nat.centralBinom
8 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA115257.a_0 line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA115257.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 OeisA115257.a_1 line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA115257.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 OeisA115257.a_2 line 49 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA115257.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 OeisA115257.a_3 line 52 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA115257.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.
def OeisA115257.polyP line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Polynomial, Rat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, RingHom, Polynomial.C, HPow.hPow, Nat.centralBinom, Polynomial.X
19 instances, coercions and auxiliary constants left out.
def OeisA115257.polyQ line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Polynomial, Rat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, RingHom, Polynomial.C, HDiv.hDiv, HPow.hPow, Nat.centralBinom, Polynomial.X
23 instances, coercions and auxiliary constants left out.
theorem OeisA115257.conjecture Documented declaration: conjecture sorry line 70 source
Source documentation
Conjecture: For any positive integer n, the polynomials Sum_{k=0}^n binomial(2k,k)^2*x^k and Sum_{k=0}^n binomial(2k,k)^2*x^k/(k+1) are irreducible over the field of rational numbers.
- Zhi-Wei Sun, Mar 23 2013
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, And, Irreducible, Polynomial, Rat, OeisA115257.polyP, OeisA115257.polyQ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 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), 1 ≤ n → Irreducible (polyP n) ∧ Irreducible (polyQ n) := by
…
```
Mentions: OeisA115257.polyP, OeisA115257.polyQ