185895.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

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

13 declarations, 3 contain sorry, 0 declared as axioms

Coefficients of $\prod_{k>0} (1 - x^k/k!)$

The sequence $a(n)$ has exponential generating function $$E(x) = \prod_{k=1}^\infty \left(1 - \frac{x^k}{k!}\right),$$ so that $a(n) = n! [x^n] \prod_{k=1}^n \left(1 - \frac{x^k}{k!}\right)$.

References:

A185895

Imports (1)

FormalConjecturesUtil

def OeisA185895.P The finite polynomial approximation $\prod_{k=1}^n (1 - X^k / k!)$ line 41 source
Source documentation
The finite polynomial approximation $\prod_{k=1}^n (1 - X^k / k!)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Polynomial, Rat
Proof uses: Finset.prod, Finset.Icc, OfNat.ofNat, HSub.hSub, HMul.hMul, RingHom, Polynomial.C, HDiv.hDiv, Nat.factorial, HPow.hPow, Polynomial.X
29 instances, coercions and auxiliary constants left out.
```
noncomputable def P (n : ℕ) : Polynomial ℚ :=
…
```
def OeisA185895.a The sequence $a(n) = n! [x^n] \prod_{k=1}^n (1 - x^k / k!)$ line 45 source
Source documentation
The sequence $a(n) = n! [x^n] \prod_{k=1}^n (1 - x^k / k!)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: ite, Eq, OfNat.ofNat, Rat.floor, HMul.hMul, Rat, Polynomial.coeff, OeisA185895.P, Nat.factorial
8 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℤ :=
…
```
def OeisA185895.IsTriangular A natural number $n$ is triangular if $n = k(k+1)/2$ for some $k \in \mathbb{N}$ line 50 source
Source documentation
A natural number $n$ is triangular if $n = k(k+1)/2$ for some $k \in \mathbb{N}$.

no axioms what this means
Statement uses: Nat
Proof uses: Exists, Eq, HDiv.hDiv, HMul.hMul, HAdd.hAdd, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.
```
def IsTriangular (n : ℕ) : Prop := ∃ k : ℕ, n = k * (k + 1) / 2
…
```
lemma OeisA185895.C_mul_X_pow_mul line 53 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Nat, Eq, Polynomial, HMul.hMul, RingHom, Polynomial.C, HPow.hPow, Polynomial.X, HAdd.hAdd
Proof uses: Eq.mpr, id, congrArg, Polynomial.C_mul, pow_add, Mathlib.Tactic.Ring.of_eq, Nat.rawCast, OfNat.ofNat, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.atom_pf, rfl, Eq.symm, Eq.refl, Mathlib.Tactic.Ring.Common.pow_congr, Mathlib.Tactic.Ring.Common.pow_add, Mathlib.Tactic.Ring.Common.single_pow, Mathlib.Tactic.Ring.Common.mul_pow_mul, Mathlib.Tactic.Ring.Common.mul_pf_right, Mathlib.Meta.NormNum.IsNat.to_raw_eq, Mathlib.Meta.NormNum.isNat_mul and 10 more
46 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem OeisA185895.a_0 Value of the sequence a at 0 line 60 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA185895.a, OfNat.ofNat, Nat
Proof uses: 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 rfl
…
```
Mentions: OeisA185895.a
theorem OeisA185895.a_1 Value of the sequence a at 1 line 64 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA185895.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: id, Eq.mpr, Rat.floor, HMul.hMul, Rat, Polynomial.coeff, Finset.prod, Polynomial, Finset.Icc, HSub.hSub, RingHom, Polynomial.C, HDiv.hDiv, Nat.factorial, HPow.hPow, Polynomial.X, congrFun', congrArg, Eq.trans, Inv.inv and 21 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = -1 := by
…
```
Mentions: OeisA185895.a
theorem OeisA185895.a_2 Value of the sequence a at 2 line 71 source
Source documentation
Value of the sequence a at 2.

extra axioms: _private.0.OeisA185895.a_2._native.decide.ax_1 what this means
Statement uses: Eq, Int, OeisA185895.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: id, Finset, Finset.Icc, Insert.insert, Singleton.singleton, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Rat.floor, HMul.hMul, Rat, Polynomial.coeff, Finset.prod, Polynomial, HSub.hSub, RingHom, Polynomial.C and 25 more
75 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = -1 := by
…
```
Mentions: OeisA185895.a
theorem OeisA185895.a_3 Value of the sequence a at 3 line 81 source
Source documentation
Value of the sequence a at 3.

extra axioms: _private.0.OeisA185895.a_3._native.decide.ax_1 what this means
Statement uses: Eq, Int, OeisA185895.a, OfNat.ofNat, Nat
Proof uses: id, Finset, Finset.Icc, Insert.insert, Singleton.singleton, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Rat.floor, HMul.hMul, Rat, Polynomial.coeff, Finset.prod, Polynomial, HSub.hSub, RingHom, Polynomial.C and 25 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 2 := by
…
```
Mentions: OeisA185895.a
theorem OeisA185895.a_4 Value of the sequence a at 4 line 91 source
Source documentation
Value of the sequence a at 4.

extra axioms: _private.0.OeisA185895.a_4._native.decide.ax_1 what this means
Statement uses: Eq, Int, OeisA185895.a, OfNat.ofNat, Nat
Proof uses: id, Finset, Finset.Icc, Insert.insert, Singleton.singleton, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, Rat.floor, HMul.hMul, Rat, Polynomial.coeff, Finset.prod, Polynomial, HSub.hSub, RingHom, Polynomial.C and 25 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 3 := by
…
```
Mentions: OeisA185895.a
def OeisA185895.c The $n$-th coefficient of the square of the ordinary generating function $A(x)^2$ line 101 source
Source documentation
The $n$-th coefficient of the square of the ordinary generating function $A(x)^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HMul.hMul, OeisA185895.a, HSub.hSub
8 instances, coercions and auxiliary constants left out.
```
noncomputable def c (n : ℕ) : ℤ :=
…
```
theorem OeisA185895.conjecture1 sorry line 111 source
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, Int, HMul.hMul, OeisA185895.a, HSub.hSub, OeisA185895.IsTriangular
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA185895.conjecture2 The coefficients $c(n)$ of $A(x)^2 = (\sum_{n \ge 0} a(n) x^n)^2$ differ in sign from $c(n-1)$ if and only if $n$ is a triangular number sorry line 121 source
Source documentation
The coefficients $c(n)$ of $A(x)^2 = (\sum_{n \ge 0} a(n) x^n)^2$ differ in sign from $c(n-1)$ if and only if $n$ is a triangular number.
- Peter Bala, Mar 17 2022
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, Int, HMul.hMul, OeisA185895.c, HSub.hSub, OeisA185895.IsTriangular
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture2 (n : ℕ) (hn : 0 < n) :
    c n * c (n - 1) < 0 ↔ IsTriangular n := by
…
```
Mentions: OeisA185895.c, OeisA185895.IsTriangular
theorem OeisA185895.conjecture3 sorry line 133 source
depends on sorry what this means
Statement uses: Nat, Nat.Prime, LT.lt, OfNat.ofNat, Int.ModEq, HPow.hPow, Int, OeisA185895.a, HMul.hMul, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

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