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import FormalConjecturesUtilEighth root of the generating function of an Apéry-like sequence
The auxiliary sequence used for the Hankel matrix, defined as $$\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$$
Determinant of the $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry equal to A005259$(i+j)$ for all $i,j = 0,\dots,n$. The entry function A005259 is taken to be $\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA228143open Polynomialopen BigOperators Matrix NatThe auxiliary sequence used for the Hankel matrix, defined as $$\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$$
def aperyLike (n : ℕ) : ℕ :=
Finset.sum (Finset.range (n + 1)) fun k =>
(n.choose k)^2 * ((Nat.choose (n + k) k))^2Determinant of the $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry equal to A005259$(i+j)$ for all $i,j = 0,\dots,n$. The entry function A005259 is taken to be $\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$.
def a (n : ℕ) : ℕ :=
let dim : Type := Fin (n + 1)
-- Matrix entries are lifted to ℤ for determinant calculation
let M : Matrix dim dim ℤ :=
Matrix.of fun i j => (aperyLike (i.val + j.val) : ℤ)
-- The sequence is known to be non-negative integers (nonn).
M.det.natAbsopen PowerSeriesThe power series $A(x/3) = \sum_{n=0}^\infty \frac{a(n)}{3^n} x^n$ over ℚ.
def ogfAScaled : PowerSeries ℚ :=
PowerSeries.mk fun n => (a n : ℚ) / (3 ^ n : ℚ)@[category test, AMS 11]
lemma a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙@[category test, AMS 11]
lemma a_1 : a 1 = 48 := ⊢ a 1 = 48 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = 161856 := ⊢ a 2 = 161856 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = 39002646528 := ⊢ a 3 = 39002646528 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = 674708032182398976 := ⊢ a 4 = 674708032182398976 All goals completed! 🐙Conjecture: if $A(x) = 1 + 48x + 161856 x^2 + \cdots$ denotes the o.g.f. then $A(x/3)^{1/8}$ has integer coefficients. - Peter Bala, Apr 22 2018
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/228143.wip.lean#L698"]
theorem exists_power_series_eighth_pow_eq : ∃ C : PowerSeries ℤ,
(PowerSeries.map (Int.castRingHom ℚ)) (C ^ 8) = ogfAScaled := ⊢ ∃ C, (PowerSeries.map (Int.castRingHom ℚ)) (C ^ 8) = ogfAScaled
All goals completed! 🐙end OeisA228143