bibkey: oeis2024a338193 authors: Vaclav Kotesovec; Mikhail Kurkov year: 2024 title: OEIS A338193, an integral-defined exponential generating function and Kurkov’s recurrence doi: null url: https://oeis.org/A338193 claim: “Conjecture: a(n) = f(0,n-1) for n > 0, where f is Kurkov’s two-index recurrence.” strata_touched:

• D5/S1/Recurrence/Algebraic/SchroderIntegralEGF license: citation-only triage: anchor

OEIS A338193

Kotesovec’s entry was created on October 15, 2020; the conjecture was added in 2024. The entry defines A by A(0)=1 and A(x)=1+Integral ((x/A(x))’ / (x/A(x)^2)’) dx, with the integral constant zero. It defines a(n) as n! times coefficient n. Kurkov’s October 26, 2024 formula conjectures a(n)=f(0,n-1) for positive n, with these three recurrences:

• f(j,0)=1 for all natural j.
• f(0,m)=f(0,m-1)+m*f(1,m-1) for positive m.
• f(j,m)=f(j-1,m)+m*(f(j,m-1)+f(j+1,m-1)) for positive j,m.

Kotesovec’s one-dimensional recurrence appears as a separate formula. The entry does not give a proof connecting Kurkov’s recurrence to the EGF. The Lean module proves that connection through the large Schroeder series, using the functional equation already in pinned Mathlib. This proof is the repository’s derivation; the source supplies the equation and conjecture.

Verified locator

url: https://oeis.org/A338193

doi: null (OEIS entry; no DOI assigned).

The full internal entry at https://oeis.org/A338193/internal was fetched and read on 2026-09-10. The definition is its %N field and the conjecture is its %F field attributed to Mikhail Kurkov, Oct 26 2024. The implementation report records the exact library searches and the limits of the literature search.