OEIS A000180: Expansion of exp(x)/(1-3x)

Join us for a deep dive into OEIS A000180: Expansion of exp(x)/(1-3x). We’ll trace its many faces—from the early terms 1, 2, 13, 16, 1,393, 20,894, 376,093 onward—to a Cloitre-style summation formula, the sharp asymptotic a(n) ~ 3^n n!/e^{1/3}, two- and three-term recurrences, and a differential equation for its exponential generating function. We’ll also connect to classic references (Riordan, Sloan, etc.) and explore related sequences that illuminate the combinatorial structure behind this intriguing sequence.

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