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A210916 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 8 for all x.

%I A210916 #15 Aug 02 2021 14:24:22
%S A210916 1,1,4,26,243,2992,45906,845287,18182926,440710385,11876274391,
%T A210916 351546957499,11330575607067,394862762014644,14792903605828298,
%U A210916 592835563146850723,25306351970600498930,1146305330627242918543,54914971513967144548105,2773947252964889935144249
%N A210916 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 8 for all x.
%C A210916 R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.
%D A210916 A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.
%H A210916 Alois P. Heinz, <a href="/A210916/b210916.txt">Table of n, a(n) for n = 0..200</a>
%F A210916 E.g.f.: t_8(x), where t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)) for k>=0 and t_k(x) = 0 otherwise.
%p A210916 t:= proc(k) option remember;
%p A210916       `if`(k<0, 0, unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
%p A210916     end:
%p A210916 gf:= t(8)(x):
%p A210916 a:= n-> n!* coeff(series(gf, x, n+1), x, n):
%p A210916 seq(a(n), n=0..30);
%t A210916 t[k_] := t[k] = If[k<0, 0, Function[x, Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]]]]; gf = t[8][x]; a[n_] := n!*SeriesCoefficient[gf, {x, 0, n}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 13 2014, translated from Maple *)
%Y A210916 Column k=8 of A135302.
%K A210916 nonn
%O A210916 0,3
%A A210916 _Alois P. Heinz_, Mar 29 2012