228143.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

Eighth root of the generating function of an Apéry-like sequence

The auxiliary sequence used for the Hankel matrix, defined as $$\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$$

Determinant of the $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry equal to A005259$(i+j)$ for all $i,j = 0,\dots,n$. The entry function A005259 is taken to be $\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$.

References:

A228143
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
A005259

Imports (1)

FormalConjecturesUtil

def OeisA228143.aperyLike The auxiliary sequence used for the Hankel matrix, defined as $$\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$$ line 50 source
Source documentation
The auxiliary sequence used for the Hankel matrix, defined as $$\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, HPow.hPow, Nat.choose
10 instances, coercions and auxiliary constants left out.
```
def aperyLike (n : ℕ) : ℕ :=
…
```
def OeisA228143.a Determinant of the $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry equal to A005259$(i+j)$ for all $i,j = 0,\dots,n$ line 59 source
Source documentation
Determinant of the $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry equal to A005259$(i+j)$ for all $i,j = 0,\dots,n$. The entry function A005259 is taken to be $\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Fin, HAdd.hAdd, OfNat.ofNat, Matrix, Int, Equiv, Matrix.of, OeisA228143.aperyLike, Int.natAbs, Matrix.det
12 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
def OeisA228143.ogfAScaled The power series $A(x/3) = \sum_{n=0}^\infty \frac{a(n)}{3^n} x^n$ over ℚ line 70 source
Source documentation
The power series $A(x/3) = \sum_{n=0}^\infty \frac{a(n)}{3^n} x^n$ over ℚ.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PowerSeries, Rat
Proof uses: PowerSeries.mk, Nat, HDiv.hDiv, OeisA228143.a, HPow.hPow, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def ogfAScaled : PowerSeries ℚ :=
…
```
lemma OeisA228143.a_0 line 75 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA228143.a, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA228143.a_1 line 78 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA228143.a, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA228143.a_2 line 81 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA228143.a, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA228143.a_3 line 84 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA228143.a, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA228143.a_4 line 87 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA228143.a, OfNat.ofNat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA228143.exists_power_series_eighth_pow_eq sorry line 98 source
depends on sorry what this means
Statement uses: Exists, PowerSeries, Int, Eq, Rat, RingHom, Unit, PowerSeries.map, Int.castRingHom, HPow.hPow, Nat, OfNat.ofNat, OeisA228143.ogfAScaled
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

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