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 A052880 Expansion of e.g.f.: LambertW(1-exp(x))/(1-exp(x)). 23
 1, 1, 4, 26, 243, 2992, 45906, 845287, 18182926, 447797646, 12429760889, 384055045002, 13075708703910, 486430792977001, 19632714343389296, 854503410602781782, 39898063449977239323, 1989371798838577172796, 105503454201101084456182, 5930110732782743218645271 (list; graph; refs; listen; history; text; internal format)
 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 Row sums of A135313. Main diagonal of A135302. Cf. A033917, A053763. Sequence in context: A210916 A210917 A210918 * A090357 A322766 A160886 Adjacent sequences: A052877 A052878 A052879 * A052881 A052882 A052883 KEYWORD easy,nonn AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 EXTENSIONS Edited by Alois P. Heinz, Nov 21 2010 STATUS approved

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Last modified June 2 14:52 EDT 2023. Contains 363097 sequences. (Running on oeis4.)