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%I #18 Nov 20 2021 10:50:19
%S 0,1,1821,141533,4798983,124878033,3068829477,75967708311,
%T 1933688266686,51075201835515,1405508547112670,40356644902123914,
%U 1209368372802130814,37806870603888974350,1231961629420423620918,41802174277488971170242,1475352032068521550599837
%N Number of transitive reflexive early confluent binary relations R on n+4 labeled elements with max_{x}(|{y : xRy}|) = n.
%C R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.
%H Alois P. Heinz, <a href="/A218104/b218104.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = A135313(n+4,n).
%F a(n) ~ n! * n^8 / (768 * log(2)^(n+5)). - _Vaclav Kotesovec_, Nov 20 2021
%p t:= proc(k) option remember; `if`(k<0, 0, unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x)) end: tt:= proc(k) option remember; unapply((t(k)-t(k-1))(x), x) end: T:= proc(n, k) option remember; coeff(series(tt(k)(x), x, n+1), x, n) *n! end:
%p a:= n-> T(n+4,n): seq(a(n), n=0..20);
%t m = 4; 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 nonn
%O 0,3
%A _Alois P. Heinz_, Oct 20 2012
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