A052807 - OEIS

%I #45 Sep 08 2024 08:00:35

%S 0,1,3,17,146,1704,25284,456224,9702776,237711888,6593032560,

%T 204212077992,6986942528400,261700394006232,10650713784774504,

%U 468007296229553880,22083086552247101184,1113646609708909274880

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

%C Previous name was: A simple grammar.

%C 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

%H G. C. Greubel, <a href="/A052807/b052807.txt">Table of n, a(n) for n = 0..375</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=769">Encyclopedia of Combinatorial Structures 769</a>

%F a(n) = Sum_{k=1..n} |Stirling1(n, k)|*k^(k-1). - _Vladeta Jovovic_, Sep 17 2003

%F E.g.f. satisfies: A(x) = 1/(1-x)^A(x). - _Paul D. Hanna_, Jul 19 2006

%F a(n) ~ n^(n-1)*exp((exp(-1)-1)*n+1/2) / (exp(exp(-1))-1)^(n-1/2). - _Vaclav Kotesovec_, Jul 09 2013

%F E.g.f.: Series_Reversion( 1 - exp(-x*exp(-x)) ). - _Seiichi Manyama_, Sep 08 2024

%e E.g.f.: A(x) = x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! +...

%e A(x)/exp(A(x)) = -log(1-x) = x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 +...

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

%t 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 *)

%t CoefficientList[Series[-LambertW[-Log[-1/(-1 + x)]], {x,0,50}], x]*

%t Range[0,50]! (* _G. C. Greubel_, Jun 18 2017 *)

%o (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

%o (PARI) x = 'x + O('x^30); concat(0, Vec(serlaplace(-lambertw(log(1-x))))) \\ _Michel Marcus_, Jun 19 2017

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

%K easy,nonn

%O 0,3

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

%E New name using e.g.f. by _Vaclav Kotesovec_, Oct 18 2013