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From the Recurrence to the Cubic

Abstract

The Catalan-weighted recurrence implies the cubic equation for the ordinary generating series.

Theorem 1.1 (The cubic from a Catalan recurrence).

Lean statement: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeAlgebra.cubic_of_recurrence

Proof. Machine-checked in Lean as D5/S3/Combinatorics/ArrowThirtyTwoOneThreeAlgebra.cubic_of_recurrence (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Robin D.P. Zhou, Xinyang Yu (2026). Arrow-Wilf equivalences and enumerative results for short arrow patterns. DOI: 10.48550/arXiv.2609.29392. URL: https://arxiv.org/abs/2609.29392v1.

Commentary.

Let a be a sequence of natural numbers with a at zero equal to one. If each positive term is its predecessor plus the Catalan-weighted sum of coefficients of powers of its generating series, then that series F satisfies 1 + (3x - 2)F + (1 - x)(1 - 2x)F squared + x cubed F cubed = 0.

References

  • Truth anchor: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeAlgebra.cubic_of_recurrence