OFFSET
0,3
COMMENTS
A simple grammar.
Also the number of transitive reflexive early confluent binary relations R on n labeled elements. Early confluency means that (xRy and xRz) implies (yRz or zRy) for all x, y, z.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..378
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 851
FORMULA
a(n) = Sum_{k=0..n} Stirling2(n, k)*(k+1)^(k-1). - Vladeta Jovovic, Nov 12 2003
a(n) ~ sqrt(1+exp(1)) * n^(n-1) / (exp(n-1)*(log(1+exp(1))-1)^(n-1/2)). - Vaclav Kotesovec, Nov 27 2012
E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} Stirling2(n,k) * A(x)^k. - Paul D. Hanna, Mar 09 2013
E.g.f. A(x) satisfies: A(x) = exp((exp(x) - 1)*A(x)). - Ilya Gutkovskiy, Apr 04 2019
MAPLE
spec := [S, {B=Set(Z, 1 <= card), S=Set(C), C=Prod(B, S)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0,
(m+1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..27); # Alois P. Heinz, Jul 15 2022
MATHEMATICA
CoefficientList[Series[-LambertW[-E^x+1]/(E^x-1), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
f[0, _] = 1; f[k_, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]];
(* b = A135302 *) b[0, 0] = 1; b[_, 0] = 0; b[n_, k_] := SeriesCoefficient[ f[k, x], {x, 0, n}]*n!;
a[n_] := b[n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 14 2019 *)
PROG
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling2(m, k)*(A+x*O(x^n))^k)*x^m/m!)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Mar 09 2013
(PARI) x='x+O('x^30); Vec(serlaplace(-lambertw(-exp(x)+1)/(exp(x)-1))) \\ G. C. Greubel, Feb 19 2018
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
Edited by Alois P. Heinz, Nov 21 2010
STATUS
approved