103885.lean`

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L4DEL: Lean4 Domain Explanation Language

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

11 declarations, 1 contain sorry, 0 declared as axioms

$a(n) = [x^{2n}] \left(\frac{1 + x}{1 - x}\right)^n$

The sequence is given by the combinatorial identity: $a(n) = \sum_{k = 0}^n \binom{n}{k} \binom{2n+k-1}{n-1}$ with $a(0) = 1$.

References:

A103885

Imports (1)

FormalConjecturesUtil

def OeisA103885.a The primary defining sequence a line 42 source
Source documentation
The primary defining sequence a. $a(n) = [x^{2n}] \left(\frac{1 + x}{1 - x}\right)^n$, given by the combinatorial identity: $a(n) = \sum_{k = 0}^n \binom{n}{k} \binom{2n+k-1}{n-1}$ with $a(0) = 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: ite, Eq, OfNat.ofNat, HSub.hSub, Finset.sum, Finset.range, HAdd.hAdd, HMul.hMul, Nat.choose
9 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA103885.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 50 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA103885.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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 a_0 : a 0 = 1 := by decide
…
```
Mentions: OeisA103885.a
theorem OeisA103885.a_1 line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA103885.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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 OeisA103885.a_2 line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA103885.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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 OeisA103885.a_3 line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA103885.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, 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 OeisA103885.a_4 line 62 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA103885.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def OeisA103885.aSubsequenceReal The sequence $b(n) = a(m*n)$ lifted to ℝ line 65 source
Source documentation
The sequence $b(n) = a(m*n)$ lifted to ℝ

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: OeisA103885.a, HMul.hMul
4 instances, coercions and auxiliary constants left out.
```
noncomputable def aSubsequenceReal (m n : ℕ) : ℝ :=
…
```
def OeisA103885.productIndices The indices $k = 1$ to $2m$, used in the product line 69 source
Source documentation
The indices $k = 1$ to $2m$, used in the product

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: Finset.Ioc, OfNat.ofNat, HMul.hMul
5 instances, coercions and auxiliary constants left out.
```
def productIndices (m : ℕ) : Finset ℕ :=
…
```
def OeisA103885.prodFactorPlus The factor $\prod_{k=1}^{2m} (2mn + k)$ line 73 source
Source documentation
The factor $\prod_{k=1}^{2m} (2mn + k)$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Finset.prod, OeisA103885.productIndices, HAdd.hAdd, HMul.hMul, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
```
noncomputable def prodFactorPlus (m n : ℕ) : ℝ :=
…
```
def OeisA103885.prodFactorMinus The factor $\prod_{k=1}^{2m} (2mn - k)$ line 78 source
Source documentation
The factor $\prod_{k=1}^{2m} (2mn - k)$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Finset.prod, OeisA103885.productIndices, HSub.hSub, HMul.hMul, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
```
noncomputable def prodFactorMinus (m n : ℕ) : ℝ :=
…
```
theorem OeisA103885.conjecture sorry line 99 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Polynomial, Real, And, Eq, WithBot, Polynomial.degree, HMul.hMul, HAdd.hAdd, OeisA103885.prodFactorPlus, Polynomial.eval, OeisA103885.aSubsequenceReal, HPow.hPow, Neg.neg, OeisA103885.prodFactorMinus, HSub.hSub, Complex and 7 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
86 instances, coercions and auxiliary constants left out.

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