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import FormalConjecturesUtilPronic indices for odd coefficients of $\sum_{n \ge 0} x^n \frac{(1+x^n)^n}{(1+x^{n+1})^{n+1}}$
Coefficients of G.f. $\sum_{n\ge 0} x^n \cdot \frac{(1 + x^n)^n}{(1 + x^{n+1})^{n+1}}$. The $m$-th term $a(m)$ is the coefficient of $x^m$, which is explicitly given by the sum: $$ a(m) = \sum_{n=0}^m \sum_{k=0}^n \binom{n}{k} (-1)^j \binom{n+j}{j},$$ where $j = \frac{m - n(k+1)}{n+1}$, and the term is zero unless $j$ is a natural number.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA323557open NatCoefficients of G.f. $\sum_{n\ge 0} x^n \cdot \frac{(1 + x^n)^n}{(1 + x^{n+1})^{n+1}}$. The $m$-th term $a(m)$ is the coefficient of $x^m$, which is explicitly given by the sum: $$ a(m) = \sum_{n=0}^m \sum_{k=0}^n \binom{n}{k} (-1)^j \binom{n+j}{j},$$ where $j = \frac{m - n(k+1)}{n+1}$, and the term is zero unless $j$ is a natural number.
def a (m : ℕ) : ℤ :=
Finset.sum (Finset.range (m + 1)) fun n =>
Finset.sum (Finset.range (n + 1)) fun k =>
let exp_x_num := n * (k + 1)
if exp_x_num ≤ m then
let remainder := m - exp_x_num
if (n + 1) ∣ remainder then
let j : ℕ := remainder / (n + 1)
let c₁ : ℤ := (n.choose k)
let c₂ : ℤ := (choose (n + j) j)
let sign : ℤ := if Even j then 1 else -1
sign * c₁ * c₂
else
0
else
0@[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 = 0 := ⊢ a 1 = 0 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = 3 := ⊢ a 2 = 3 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = -2 := ⊢ a 3 = -2 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = 2 := ⊢ a 4 = 2 All goals completed! 🐙Conjecture: Odd terms occur only at positions $n(n+1)$ for $n \ge 0$ (verified for initial 32600 terms).
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/323557.wip.lean#L193"]
theorem odd_a_implies_pronic (m : ℕ) : Odd (a m) → ∃ n : ℕ, m = n * (n + 1) := m:ℕ⊢ Odd (a m) → ∃ n, m = n * (n + 1)
All goals completed! 🐙end OeisA323557