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import FormalConjecturesUtilNumerator of $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right)$
Conjecture: $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right) = A005156(n+1)/A005156(n)$
References:
namespace OeisA109074open NatThe rational number defined by $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right)$, whose numerator is A109074.
def frac (n : ℕ) : ℚ :=
let numTerm : ℕ := (6 * n - 2).choose (2 * n)
let denTerm : ℕ := 2 * ((4 * n - 1).choose (2 * n))
(numTerm : ℚ) / (denTerm : ℚ)
The primary defining sequence a.
$a(n)$ is the numerator of $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right)$.
def a (n : ℕ) : ℕ :=
(frac n).num.natAbs@[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 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 3 := ⊢ a 2 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 26 := ⊢ a 3 = 26 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 323 := ⊢ a 4 = 323 All goals completed! 🐙A005156: The sequence of values $\frac{1}{2n+1} \binom{3n}{n}$.
def b (n : ℕ) : ℕ :=
let num : ℕ := (3 * n).choose n
let den : ℕ := 2 * n + 1
num / denIt is conjectured that binomial(6n-2,2n)/(2 * binomial(4n-1,2n)) = A005156(n+1)/A005156(n).
@[category research open, AMS 11]
theorem conjecture (n : ℕ) (h_pos : n ≥ 1) :
frac n = (b (n + 1) : ℚ) / (b n : ℚ) := n:ℕh_pos:n ≥ 1⊢ frac n = ↑(b (n + 1)) / ↑(b n)
All goals completed! 🐙end OeisA109074