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import FormalConjecturesUtilIntegrality of the factorial ratio $\frac{(12n+6)! (6n+9)!}{108 (4n+2)! (2n+3)! ((6n+5)!)^2}$
The factorial ratio $$a(n) = \frac{(12n + 6)! \cdot (6n + 9)!}{108 \cdot (4n + 2)! \cdot (2n + 3)! \cdot ((6n + 5)!)^2}$$ It is conjectured that $a(n)$ are integers.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA368692open NatThe factorial ratio $$a(n) = \frac{(12n + 6)! \cdot (6n + 9)!}{108 \cdot (4n + 2)! \cdot (2n + 3)! \cdot ((6n + 5)!)^2}$$ It is conjectured that $a(n)$ are integers.
def a (n : ℕ) : ℕ :=
let num : ℕ := (12 * n + 6)! * (6 * n + 9)!
let den_base : ℕ := (4 * n + 2)! * (2 * n + 3)! * ((6 * n + 5)!)^2
num / (108 * den_base)@[category test, AMS 11]
lemma a_0 : a 0 = 14 := ⊢ a 0 = 14 All goals completed! 🐙@[category test, AMS 11]
lemma a_1 : a 1 = 563108 := ⊢ a 1 = 563108 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = 54231252075 := ⊢ a 2 = 54231252075 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = 6700034035890000 := ⊢ a 3 = 6700034035890000 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = 928978310614152999200 := ⊢ a 4 = 928978310614152999200 All goals completed! 🐙It is conjectured here that $a(n)$ are integers.
A formal proof has been found with the methods described in arxiv/2605.22763.
@[category research solved, AMS 11, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/368692.wip.lean#L158"]
theorem a_is_int (n : ℕ) : 108 * ((4 * n + 2)! * (2 * n + 3)! * ((6 * n + 5)!) ^ 2) ∣
(12 * n + 6)! * (6 * n + 9)! := n:ℕ⊢ 108 * ((4 * n + 2)! * (2 * n + 3)! * (6 * n + 5)! ^ 2) ∣ (12 * n + 6)! * (6 * n + 9)!
All goals completed! 🐙end OeisA368692