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import FormalConjecturesUtilCentral factorial numbers: $((2n)!!)^2$
Central factorial numbers: $a(n) = 4^n (n!)^2 = ((2n)!!)^2$.
References:
namespace OeisA2454Central factorial numbers: $a(n) = 4^n (n!)^2$.
def a (n : ℕ) : ℕ :=
4 ^ n * n.factorial ^ 2
Value of the sequence a at 0.
@[category test, AMS 11]
theorem a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 4 := ⊢ a 1 = 4 All goals completed! 🐙
Value of the sequence a at 2.
@[category test, AMS 11]
theorem a_2 : a 2 = 64 := ⊢ a 2 = 64 All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 2304 := ⊢ a 3 = 2304 All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 147456 := ⊢ a 4 = 147456 All goals completed! 🐙Let $\zeta$ be a primitive $(2n+1)$-th root of unity. Then the permanent of the $2n \times 2n$ matrix $[m(j,k)]_{j,k=1..2n}$ is $a(n)/(2n+1) = ((2n)!!)^2/(2n+1)$, where $m(j,k)$ is $1$ or $(1+\zeta^{j-k})/(1-\zeta^{j-k})$ according as $j = k$ or not.
Zhi-Wei Sun, Dec 21 2021
@[category research open, AMS 11 15]
theorem conjecture (n : ℕ) :
let N : ℕ := 2 * n
let K : ℕ := N + 1
∀ (ζ : ℂ), IsPrimitiveRoot ζ K →
Matrix.permanent (fun (j k : Fin N) =>
if j = k then
(1 : ℂ)
else
let pow : ℤ := (j : ℤ) - (k : ℤ)
(1 + ζ ^ pow) / (1 - ζ ^ pow)
) = (a n : ℂ) / (K : ℂ) := n:ℕ⊢ let N := 2 * n;
let K := N + 1;
∀ (ζ : ℂ),
IsPrimitiveRoot ζ K →
(Matrix.permanent fun j k ↦
if j = k then 1
else
let pow := ↑↑j - ↑↑k;
(1 + ζ ^ pow) / (1 - ζ ^ pow)) =
↑(OeisA2454.a n) / ↑K
All goals completed! 🐙end OeisA2454