A218108 - OEIS

A218108

Number of transitive reflexive early confluent binary relations R on n+8 labeled elements with max_{x}(|{y : xRy}|) = n.

%I #13 Aug 02 2021 08:34:50

%S 0,1,4910785,6135529675,1008618127825,74564772630777,3913397076494906,

%T 177779119899659850,7631859877504516225,322215964319093498225,

%U 13636766011245325587353,584294217357391235758011,25488316708898114509899955,1135731969645865474902932115

%N Number of transitive reflexive early confluent binary relations R on n+8 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="/A218108/b218108.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = A135313(n+8,n).

%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+8,n): seq (a(n), n=0..20);

%t m = 8; 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