A052880 Expansion of e.g.f.: LambertW(1-exp(x))/(1-exp(x)).

1, 1, 4, 26, 243, 2992, 45906, 845287, 18182926, 447797646, 12429760889, 384055045002, 13075708703910, 486430792977001, 19632714343389296, 854503410602781782, 39898063449977239323, 1989371798838577172796, 105503454201101084456182, 5930110732782743218645271

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

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

Crossrefs

Row sums of A135313.

Main diagonal of A135302.

Cf. A033917, A053763.

Programs

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 *)

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

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

Extensions

Edited by Alois P. Heinz, Nov 21 2010