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import FormalConjecturesUtil
namespace OeisA115257open Polynomial
The primary defining sequence a.
Partial sums of $\binom{2n}{n}^2$.
def a (n : ℕ) : ℕ :=
(Finset.range (n + 1)).sum (fun k => (Nat.centralBinom k) ^ 2)@[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 = 5 := ⊢ a 1 = 5 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 41 := ⊢ a 2 = 41 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 441 := ⊢ a 3 = 441 All goals completed! 🐙The polynomial $\sum_{k=0}^{n} \binom{2k}{k}^2 x^k$ over $\mathbb{Q}$.
noncomputable
def polyP (n : ℕ) : Polynomial ℚ :=
(Finset.range (n + 1)).sum (fun k => C ((Nat.centralBinom k : ℚ) ^ 2) * X ^ k)The polynomial $\sum_{k=0}^{n} \frac{\binom{2k}{k}^2}{k+1} x^k$ over $\mathbb{Q}$.
noncomputable
def polyQ (n : ℕ) : Polynomial ℚ :=
(Finset.range (n + 1)).sum (fun k => C (((Nat.centralBinom k : ℚ) ^ 2) / (k + 1 : ℚ)) * X ^ k)Conjecture: For any positive integer n, the polynomials Sum_{k=0}^n binomial(2k,k)^2x^k and Sum_{k=0}^n binomial(2k,k)^2x^k/(k+1) are irreducible over the field of rational numbers.
Zhi-Wei Sun, Mar 23 2013
@[category research open, AMS 11]
theorem conjecture :
∀ (n : ℕ), 1 ≤ n → Irreducible (polyP n) ∧ Irreducible (polyQ n) := ⊢ ∀ (n : ℕ), 1 ≤ n → Irreducible (polyP n) ∧ Irreducible (polyQ n)
All goals completed! 🐙end OeisA115257