OFFSET
0,3
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
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..375
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 769
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
EXAMPLE
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 +...
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
New name using e.g.f. by Vaclav Kotesovec, Oct 18 2013
STATUS
approved