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A210913 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 5 for all x.

%I A210913 #16 Aug 02 2021 08:00:36
%S A210913 1,1,4,26,243,2992,41223,660220,11979669,243048992,5448497434,
%T A210913 133595966164,3555887814602,102064563003898,3141580135645330,
%U A210913 103198691666336823,3602725068242695657,133174089439019869924,5195463138498447345478,213295995976349091757666
%N A210913 Number of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= 5 for all x.
%C A210913 R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.
%D A210913 A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.
%H A210913 Alois P. Heinz, <a href="/A210913/b210913.txt">Table of n, a(n) for n = 0..200</a>
%F A210913 E.g.f.: t_5(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 A210913 t:= proc(k) option remember;
%p A210913       `if`(k<0, 0, unapply(exp(add(x^m/m!*t(k-m)(x), m=1..k)), x))
%p A210913     end:
%p A210913 gf:= t(5)(x):
%p A210913 a:= n-> n!*coeff(series(gf, x, n+1), x, n):
%p A210913 seq(a(n), n=0..30);
%t A210913 t[0, _] = 1; t[k_, x_] := t[k, x] = Exp[Sum[x^m/m!*t[k - m, x], {m, 1, k}]]; a[0, 0] = 1; a[_, 0] = 0; a[n_, k_] := SeriesCoefficient[t[k, x], {x, 0, n}]*n!; Table[a[n, 5], {n, 0, 30} ] (* _Jean-François Alcover_, Feb 04 2014, after A135302 and _Alois P. Heinz_ *)
%Y A210913 Column k=5 of A135302.
%K A210913 nonn
%O A210913 0,3
%A A210913 _Alois P. Heinz_, Mar 29 2012