A052807 Expansion of e.g.f. -LambertW(log(1-x)).

0, 1, 3, 17, 146, 1704, 25284, 456224, 9702776, 237711888, 6593032560, 204212077992, 6986942528400, 261700394006232, 10650713784774504, 468007296229553880, 22083086552247101184, 1113646609708909274880

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

E.g.f. of A052813 equals exp(A(x)) = -A(x)/log(1-x). a(n) = n!*Sum_{k=0..n-1} A052813(k)/k!/(n-k). - Paul D. Hanna, Jul 19 2006

Examples

			E.g.f.: A(x) = x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! +...
A(x)/exp(A(x)) = -log(1-x) = x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..375
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 769

Crossrefs

Cf. A006963, A048802, A052813 (exp(A(x))), A277489.

Programs

Maple

spec := [S,{B=Cycle(Z),C=Set(S),S=Prod(C,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

max = 17; se = Series[-ProductLog[-Log[-(-1 + x)^(-1)]] , {x, 0, max}]; Join[{0}, (CoefficientList[se, x] // DeleteCases[#, 0] &) * Range[max]!] (* Jean-François Alcover, Jun 24 2013 *)
CoefficientList[Series[-LambertW[-Log[-1/(-1 + x)]], {x,0,50}], x]*
Range[0,50]! (* G. C. Greubel, Jun 18 2017 *)

PARI

{a(n)=local(A=1+x);for(i=1,n,A=1/(1-x+x*O(x^n))^A);n!*polcoeff(log(A),n)} \\ Paul D. Hanna, Jul 19 2006

PARI

x = 'x + O('x^30); concat(0, Vec(serlaplace(-lambertw(log(1-x))))) \\ Michel Marcus, Jun 19 2017

Formula

a(n) = Sum_{k=1..n} |Stirling1(n, k)|*k^(k-1). - Vladeta Jovovic, Sep 17 2003

E.g.f. satisfies: A(x) = 1/(1-x)^A(x). - Paul D. Hanna, Jul 19 2006

a(n) ~ n^(n-1)*exp((exp(-1)-1)*n+1/2) / (exp(exp(-1))-1)^(n-1/2). - Vaclav Kotesovec, Jul 09 2013

E.g.f.: Series_Reversion( 1 - exp(-x*exp(-x)) ). - Seiichi Manyama, Sep 08 2024

Extensions

New name using e.g.f. by Vaclav Kotesovec, Oct 18 2013