FormalConjectures/OEIS/278070.lean

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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

Congruences $a(n+k) \equiv a(n) \pmod k$ for hypergeometric values ${}_2F_1(n, -n; ; -1)$

a: $a(n) = \text{hypergeometric}([n, -n], [], -1)$. This is equivalent to the combinatorial sum: $$a(n) = \sum_{k=0}^n \binom{n}{k} \binom{n+k-1}{k} k!$$ The expression uses $\mathbb{N}$ arithmetic throughout, safely handling the subtraction via Nat.pred.

References:

Imports (1)
  1. def OeisA278070.a a: $a(n) = \text{hypergeometric}([n, -n], [], -1)$ line 48 source

    Source documentation

    a: $a(n) = \text{hypergeometric}([n, -n], [], -1)$. This is equivalent to the combinatorial sum: $$a(n) = \sum_{k=0}^n \binom{n}{k} \binom{n+k-1}{k} k!$$ The expression uses $\mathbb{N}$ arithmetic throughout, safely handling the subtraction via Nat.pred.

    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, Nat.choose, Nat.pred, Nat.factorial

    6 instances, coercions and auxiliary constants left out.

    def a (n : ℕ) : ℕ :=
…
  2. lemma OeisA278070.a_0 line 54 source

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

    Statement uses: Eq, Nat, OeisA278070.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 OeisA278070.a_1 line 57 source

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

    Statement uses: Eq, Nat, OeisA278070.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 OeisA278070.a_2 line 60 source

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

    Statement uses: Eq, Nat, OeisA278070.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 OeisA278070.a_3 line 63 source

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

    Statement uses: Eq, Nat, OeisA278070.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 OeisA278070.a_4 line 66 source

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

    Statement uses: Eq, Nat, OeisA278070.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 OeisA278070.a_modeq sorry line 77 source

    depends on sorry what this means

    Statement uses: Nat, Nat.ModEq, OeisA278070.a, HAdd.hAdd

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

    5 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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