OFFSET
0,3
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
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
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 20 2012
STATUS
approved