FormalConjectures/OEIS/325046.lean

GitHub source file

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL
Lean4 Domain Explanation Language

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 term $a(N)$ is the coefficient of $x^N$ in the generating function. Expanding the terms, we get a formula for $a(N)$: $$a(N) = \sum_{n=0}^N \sum_{k=0}^n \mathbf{1}_{n + nk + (n+1)j = N} \binom{n}{k} \binom{n+j}{j}$$ where $j = \frac{N - n(k+1)}{n+1}$.

References:

Imports (1)
  1. def OeisA325046.a Coefficients of G.f line 52 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 term $a(N)$ is the coefficient of $x^N$ in the generating function. Expanding the terms, we get a formula for $a(N)$: $$a(N) = \sum_{n=0}^N \sum_{k=0}^n \mathbf{1}_{n + nk + (n+1)j = N} \binom{n}{k} \binom{n+j}{j}$$ where $j = \frac{N - n(k+1)}{n+1}$.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat

    Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HSub.hSub, HMul.hMul, ite, And, LE.le, Eq, HMod.hMod, HDiv.hDiv, Nat.choose

    16 instances, coercions and auxiliary constants left out.

    def a (N : ℕ) : ℕ :=
…
  2. lemma OeisA325046.a_0 line 72 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA325046.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.

  3. lemma OeisA325046.a_1 line 75 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA325046.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.

  4. lemma OeisA325046.a_2 line 78 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA325046.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.

  5. lemma OeisA325046.a_3 line 81 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA325046.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.

  6. lemma OeisA325046.a_4 line 84 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA325046.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.

  7. theorem OeisA325046.odd_a_implies_pronic sorry line 99 source

    depends on sorry what this means

    Statement uses: Nat, Eq, HMod.hMod, OeisA325046.a, OfNat.ofNat, Exists, HMul.hMul, HAdd.hAdd

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    14 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.