formal-conjectures · oeis · ams-11 · ams-15
Squares of double factorials: $a(n) = ((2n-1)!!)^2 = (1 \cdot 3 \cdot 5 \cdots (2n-1))^2$.
Conjecture 1: For any primitive $2n$-th root $\zeta$ of unity, the permanent of the $2n \times 2n$
matrix $[m(j,k)]_{j,k=1..2n}$ coincides with $a(n) = ((2n-1)!!)^2$, where $m(j,k)$ is
$(1+\zeta^{j-k})/(1-\zeta^{j-k})$ if $j \neq k$, and $1$ otherwise.
- Zhi-Wei Sun, Dec 21 2021
Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).
Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.
Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A001818
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/1818.lean
Local target: Corpus.OEIS1818.conjecture1
Source SHA-256: 3d538aeaa99e17b8c2b882912155a96ebeaeb515b8d6e1729632e7ed720efebd
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.
Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.
Conjecture 1: For any primitive $2n$-th root $\zeta$ of unity, the permanent of the $2n \times 2n$ matrix $[m(j,k)]_{j,k=1..2n}$ coincides with $a(n) = ((2n-1)!!)^2$, where $m(j,k)$ is $(1+\zeta^{j-k})/(1-\zeta^{j-k})$ if $j \neq k$, and $1$ otherwise.
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