A366672 a(n) = A002720(n)^n.

1, 2, 49, 39304, 1908029761, 8831763846882976, 5602661527604399327549089, 659308109505417338723017914068713088, 18666765602783048904120522995911258148623099215361, 159740893387079678500933964995221201596055121928224632284394525184

Offset: 0

Author

Paulius Dilkas, Oct 15 2023

Keywords

• nonn
• easy

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).

Examples

			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.
		

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.

Crossrefs

Cf. A002720, A036740.

Programs

Mathematica

Table[(n!*LaguerreL[n, -1])^n, {n, 0, 10}] (* Vaclav Kotesovec, Oct 20 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