A360820 a(n) = Sum_{k=0..n} binomial(n, k)*2^(n^2-k*(n-k)).

1, 4, 48, 1792, 221184, 98566144, 173946175488, 1281755680079872, 39534286378918477824, 5018464395368794081460224, 2586745980900067184722499862528, 5375203895735606878055792019528220672, 44865714160227204455469409035569750630989824, 1501355804811017489524770237231795462175548447391744

Offset: 0

Author

Paulius Dilkas, Feb 21 2023

Keywords

• nonn
• easy

Comments

This is the model count of the following sentence in first-order logic: forall X. forall Y. friends(X, Y) /\ smokes(X) -> smokes(Y).

Examples

			If the domain is {1}, then all four interpretations ({}, {smokes(1)}, {friends(1, 1)}, {smokes(1), friends(1, 1)}) are models, so a(1) = 4.
		

Links

• G. Van den Broeck, W. Meert, and A. Darwiche, Skolemization for weighted first-order model counting, arXiv:1312.5378 [cs.AI], 2013-2014.

Programs

Mathematica

a[n_] := Sum[Binomial[n, k]*2^(n^2 - k*(n - k)), {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 24 2023 *)

PARI

a(n) = sum(k=0, n, binomial(n, k)*2^(n^2-k*(n-k))); \\ Michel Marcus, Feb 22 2023

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*2^(n^2-k*(n-k)).

a(n) ~ 2^(n^2 + 1). - Vaclav Kotesovec, Feb 22 2023