bibkey: hanna2025a386655a386658
authors: Paul D. Hanna
year: 2025
title: OEIS A386655–A386658, Lambert W exponential generating functions
doi: null
url: https://oeis.org/A386655
claim: "Conjecture: for n >= 6, a(n) (mod 6) equals [1, 0, 3, 0, 3, 2] repeating."
strata_touched:
  - D5/S3/Combinatorics/HannaLambertW
license: citation-only
triage: anchor

Hanna’s Lambert W coefficient conjectures

Verified locator

Source: https://oeis.org/A386655

The NAME of A386655 is “E.g.f.: Sum_{n>=0} (2^nx + LambertW(x))^n / n!.” Its COMMENTS sentence is “Conjecture: for n >= 6, a(n) (mod 6) equals [1, 0, 3, 0, 3, 2] repeating.” Its FORMULA sentence is “a(n) = Sum_{k=0..n} binomial(n,k) * 2^(k(k+1)) * (2^k - (n-k))^(n-k-1).” OEIS is unpaginated; these locators refer to the NAME and COMMENTS fields.

The neighbouring entries https://oeis.org/A386656, https://oeis.org/A386657, and https://oeis.org/A386658 have the respective NAME sentences:

• “E.g.f.: Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.”
• “E.g.f.: Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.”
• “E.g.f.: Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.”

Their respective COMMENTS sentences are:

• “Conjecture: for n >= 6, a(n) (mod 6) equals [4, 3, 2, 1, 0, 1] repeating.”
• “Conjecture: for n >= 6, a(n) (mod 6) equals [1, 2, 3, 2, 3, 4] repeating.”
• “Conjecture: for n >= 6, a(n) (mod 6) equals [4, 3, 0, 3, 0, 5] repeating.”

The Lean definitions use the principal formal Lambert W series with coefficient (-m)^(m-1)/m! in positive degree m and constant coefficient zero. The coefficient function is m! times the finite coefficient sum over n at most m. The congruence includes integrality and uses zero-based pattern indexing.

The exponential-of-W coefficient identity is classical Lagrange inversion; no novelty is asserted for that identity. Literature-search completeness for the four congruence conjectures is ASSUMED-UNVERIFIED.