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A366672
a(n) = A002720(n)^n.
0
1, 2, 49, 39304, 1908029761, 8831763846882976, 5602661527604399327549089, 659308109505417338723017914068713088, 18666765602783048904120522995911258148623099215361, 159740893387079678500933964995221201596055121928224632284394525184
OFFSET
0,2
COMMENTS
This is the model count of the following sentence in first-order logic:
(forall w, x, y, z. P(x, y, z) /\ P(w, y, z) => x = w) /\
(forall w, x, y, z. P(x, y, z) /\ P(x, w, z) => y = w).
LINKS
Paulius Dilkas and Vaishak Belle, Synthesising Recursive Functions for First-Order Model Counting: Challenges, Progress, and Conjectures, KR 2023, 198-207; arXiv:2306.04189 [cs.LO], 2023.
FORMULA
a(n) ~ n^(n*(n + 1/4)) / (2^(n/2) * exp(n^2 - 2*n^(3/2) + n/2 - 31*sqrt(n)/48 + 17/192)) * (1 - 281/(5120*sqrt(n)) + 3074161/(52428800*n)). - Vaclav Kotesovec, Oct 20 2023
EXAMPLE
When n = 2, i.e., the domain is [2] = {1, 2}, both P(x, y, 1) and P(x, y, 2) represent partial injective functions from [2] to [2]. Since there are seven such functions, a(n) = 7^2 = 49.
MATHEMATICA
Table[(n!*LaguerreL[n, -1])^n, {n, 0, 10}] (* Vaclav Kotesovec, Oct 20 2023 *)
CROSSREFS
Sequence in context: A269839 A088067 A145676 * A335406 A291188 A225795
KEYWORD
nonn,easy
AUTHOR
Paulius Dilkas, Oct 15 2023
STATUS
approved