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import FormalConjecturesUtilCongruences $a(n+k) \equiv a(n) \pmod k$ for hypergeometric values ${}_2F_1(n, -n; ; -1)$
a: $a(n) = \text{hypergeometric}([n, -n], [], -1)$.
This is equivalent to the combinatorial sum:
$$a(n) = \sum_{k=0}^n \binom{n}{k} \binom{n+k-1}{k} k!$$
The expression uses $\mathbb{N}$ arithmetic throughout, safely handling
the subtraction via Nat.pred.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA278070open Nat Finset
a: $a(n) = \text{hypergeometric}([n, -n], [], -1)$.
This is equivalent to the combinatorial sum:
$$a(n) = \sum_{k=0}^n \binom{n}{k} \binom{n+k-1}{k} k!$$
The expression uses $\mathbb{N}$ arithmetic throughout, safely handling
the subtraction via Nat.pred.
def a (n : ℕ) : ℕ :=
(Finset.range (n + 1)).sum fun k =>
(n.choose k) * ((n + k).pred.choose k) * (k.factorial)@[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 = 2 := ⊢ a 1 = 2 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = 11 := ⊢ a 2 = 11 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = 106 := ⊢ a 3 = 106 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = 1457 := ⊢ a 4 = 1457 All goals completed! 🐙Conjecture: $a(n+k) \equiv a(n) \pmod{k}$ for all $n$ and $k$. - Peter Bala, Mar 12 2023
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/278070.wip.lean#L68"]
theorem a_modeq : ∀ (n k : ℕ), Nat.ModEq k (a (n + k)) (a n) := ⊢ ∀ (n k : ℕ), a (n + k) ≡ a n [MOD k]
All goals completed! 🐙end OeisA278070