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import FormalConjecturesUtilApéry numbers
Apéry numbers: $$a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}$$
References:
namespace OeisA5258Apéry numbers: $a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}$.
def a (n : ℕ) : ℕ :=
∑ k ∈ Finset.range (n + 1), n.choose k ^ 2 * (n + k).choose k@[category test, AMS 11]
theorem a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 3 := ⊢ a 1 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 19 := ⊢ a 2 = 19 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 147 := ⊢ a 3 = 147 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 1251 := ⊢ a 4 = 1251 All goals completed! 🐙open Polynomial inThe polynomial associated with the $n$-th Apéry number: $a_n(x) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} x^k$.
noncomputable def aperyPoly (n : ℕ) : ℚ[X] :=
∑ k ∈ Finset.range (n + 1),
C (((n.choose k) ^ 2 * ((n + k).choose k) : ℕ) : ℚ) * X ^ kFor each $n = 1, 2, 3, \dots$ the polynomial $a_n(x) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} x^k$ is irreducible over the field of rational numbers.
Zhi-Wei Sun, Mar 21 2013
@[category research open, AMS 11 12]
theorem conjecture (n : ℕ) (hn : 1 ≤ n) : Irreducible (aperyPoly n) := n:ℕhn:1 ≤ n⊢ Irreducible (aperyPoly n)
All goals completed! 🐙end OeisA5258