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Number of connected irreducible quasiorders with n labeled points.
1

%I #23 Jan 04 2025 02:42:16

%S 1,1,1,1,25,1321,70201,4542721,384969649,44087846545,6926924885881,

%T 1503058888234201,451117640363382697,186980881340749198561,

%U 106678398214255092939169,83440038893764124092029601,89093417035281194970121062073,129323858612953057624127147727913,254190262374139251098507525465587609

%N Number of connected irreducible quasiorders with n labeled points.

%H John A. Wright, <a href="https://www.ams.org/journals/notices/197006/197006FullIssue.pdf">There are 718 6-point topologies, quasi-orderings and transgraphs</a>, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.

%H John A. Wright, <a href="/A000798/a000798_4.pdf">Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences</a>

%F From _Geoffrey Critzer_, Jul 10 2022: (Start)

%F E.g.f.: 1 + log(A(x)) - A(x)*(1 - 1/A(x))^2 where A(x) is the e.g.f. for A000798.

%F a(n) = A001929(n) - Sum_{k>=2} A335987(n,k). (End)

%t nn = 18; A[x_] :=Total[Cases[Import["https://oeis.org/A000798/b000798.txt",

%t "Table"], {_, _}][[All, 2]]*Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]];

%t Range[0, nn]! CoefficientList[Series[1 + Log[A[x]] - A[x] (1 - 1/A[x])^2, {x, 0, nn}], x] (* _Geoffrey Critzer_, Jul 10 2022 *)

%Y Cf. A046909, A335987, A000798, A001929.

%K nonn

%O 0,5

%A John A. Wright

%E a(8)-a(18) from _Geoffrey Critzer_, Jul 10 2022