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

0, 1, 310, 12075, 267715, 5287506, 105494886, 2185028130, 47488375440, 1087116745385, 26234041133443, 666937354457829, 17839235553096685, 501241620987647540, 14769149279798467900, 455566464561064320948, 14685947990441112405726, 493969048893703131221475

Offset: 0

Author

Alois P. Heinz, Oct 20 2012

Keywords

• nonn

Comments

R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Programs

Maple

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:
a:= n-> T(n+3,n):
seq(a(n), n=0..20);

Mathematica

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 *)

Formula

a(n) = A135313(n+3,n).

a(n) ~ n! * n^6 / (96 * log(2)^(n+4)). - Vaclav Kotesovec, Nov 20 2021

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