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A218103 Number of transitive reflexive early confluent binary relations R on n+3 labeled elements with max_{x}(|{y : xRy}|) = n.

%I A218103 #23 Nov 20 2021 10:52:14
%S A218103 0,1,310,12075,267715,5287506,105494886,2185028130,47488375440,
%T A218103 1087116745385,26234041133443,666937354457829,17839235553096685,
%U A218103 501241620987647540,14769149279798467900,455566464561064320948,14685947990441112405726,493969048893703131221475
%N A218103 Number of transitive reflexive early confluent binary relations R on n+3 labeled elements with max_{x}(|{y : xRy}|) = n.
%C A218103 R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.
%H A218103 Alois P. Heinz, <a href="/A218103/b218103.txt">Table of n, a(n) for n = 0..200</a>
%F A218103 a(n) = A135313(n+3,n).
%F A218103 a(n) ~ n! * n^6  / (96 * log(2)^(n+4)). - _Vaclav Kotesovec_, Nov 20 2021
%F A218103 Conjecture: For fixed k>=0, A135313(n+k,n) ~ n! * n^(2*k)  / (2^(k+1) * k! * log(2)^(n+k+1)). - _Vaclav Kotesovec_, Nov 20 2021
%p A218103 t:= proc(k) option remember; `if`(k<0, 0,
%p A218103       unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
%p A218103     end:
%p A218103 tt:= proc(k) option remember; unapply((t(k)-t(k-1))(x), x) end:
%p A218103 T:= proc(n, k) option remember;
%p A218103       coeff(series(tt(k)(x), x, n+1), x, n) *n!
%p A218103     end:
%p A218103 a:= n-> T(n+3,n):
%p A218103 seq(a(n), n=0..20);
%t A218103 m = 3; f[0, _] = 1; f[k_, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]]; (* t = A135302 *) t[0, 0] = 1; t[_, 0] = 0; t[n_, k_] := t[n, k] = SeriesCoefficient[f[k, x], {x, 0, n}]*n!; a[0] = 0; a[n_] := t[n+m, n]-t[n+m, n-1]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 14 2014 *)
%K A218103 nonn
%O A218103 0,3
%A A218103 _Alois P. Heinz_, Oct 20 2012