This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A333331 #38 Mar 23 2024 08:29:14
%S A333331 1,3,17,144,1623,22804,383415,7501422
%N A333331 Number of integer points in the convex hull in R^n of parking functions of length n.
%C A333331 It is observed by _Gus Wiseman_ in A368596 and A368730 that this sequence appears to be the complement of those sequences. If this is the case, then a(n) is the number of labeled graphs with loops allowed in which each connected component has an equal number of vertices and edges and the conjectured formula holds. Terms for n >= 9 are expected to be 167341283, 4191140394, 116425416531, ... - _Andrew Howroyd_, Jan 10 2024
%C A333331 From _Gus Wiseman_, Mar 22 2024: (Start)
%C A333331 An equivalent conjecture is that a(n) is the number of loop-graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. I call these graphs choosable. For example, the a(3) = 17 choosable loop-graphs are the following (loops shown as singletons):
%C A333331 {{1},{2},{3}} {{1},{2},{1,3}} {{1},{1,2},{1,3}} {{1,2},{1,3},{2,3}}
%C A333331 {{1},{2},{2,3}} {{1},{1,2},{2,3}}
%C A333331 {{1},{3},{1,2}} {{1},{1,3},{2,3}}
%C A333331 {{1},{3},{2,3}} {{2},{1,2},{1,3}}
%C A333331 {{2},{3},{1,2}} {{2},{1,2},{2,3}}
%C A333331 {{2},{3},{1,3}} {{2},{1,3},{2,3}}
%C A333331 {{3},{1,2},{1,3}}
%C A333331 {{3},{1,2},{2,3}}
%C A333331 {{3},{1,3},{2,3}}
%C A333331 (End)
%D A333331 R. P. Stanley (Proposer), Problem 12191, Amer. Math. Monthly, 127:6 (2020), 563.
%H A333331 Thomas Selig, <a href="https://arxiv.org/abs/2209.07301">The stochastic sandpile model on complete graphs</a>, arXiv:2209.07301 [math.PR], 2022.
%F A333331 Conjectured e.g.f.: exp(-log(1-T(x))/2 + T(x)/2 - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - _Andrew Howroyd_, Jan 10 2024
%e A333331 For n=2 the parking functions are (1,1), (1,2), (2,1). They are the only integer points in their convex hull. For n=3, in addition to the 16 parking functions, there is the additional point (2,2,2).
%Y A333331 All of the following relative references pertain to the conjecture:
%Y A333331 The case of unique choice A000272.
%Y A333331 The version without the choice condition is A014068, covering A368597.
%Y A333331 The case of just pairs A137916.
%Y A333331 For any number of edges of any positive size we have A367902.
%Y A333331 The complement A368596, covering A368730.
%Y A333331 Allowing edges of any positive size gives A368601, complement A368600.
%Y A333331 Counting by singletons gives A368924.
%Y A333331 For any number of edges we have A368927, complement A369141.
%Y A333331 The connected case is A368951.
%Y A333331 The unlabeled version is A368984, complement A368835.
%Y A333331 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A333331 A006125 counts graphs, unlabeled A000088.
%Y A333331 A058891 counts set-systems (without singletons A016031), unlabeled A000612.
%Y A333331 Cf. A000169, A000666, A057500, A062740, A129271, A133686, A367863, A367903, A369146, A369199.
%K A333331 nonn,more
%O A333331 1,2
%A A333331 _Richard Stanley_, Mar 15 2020