%I #27 Jul 04 2018 05:44:10

%S 0,1,2,15,188,3300,74484,2054864,66998448,2520581400,107472778320,

%T 5121576763512,269759385873504,15561785854196400,975788232119245440,

%U 66080957140527828480,4806533577745476290304,373724762062131412853760

%N E.g.f. equals the series reversion of x + x*log(1-x).

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=784">Encyclopedia of Combinatorial Structures 784</a>

%H Vladimir Kruchinin, <a href="https://arxiv.org/abs/1211.3244">The method for obtaining expressions for coefficients of reverse generating functions</a>, arXiv:1211.3244 [math.CO], 2012.

%F E.g.f. satisfies: A(x + x*log(1-x)) = x. - _Paul D. Hanna_, Aug 28 2008

%F E.g.f. A(x) satisfies [from _Paul D. Hanna_, Jul 15 2012]:

%F (1) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*(-log(1-x))^n/n!.

%F (2) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*(-log(1-x))^n/n! ).

%F a(n) = Sum_{k=0..n-1} k!*(-1)^(n+k-1)*Stirling1(n-1,k)*binomial(n+k-1,n-1). - _Vladimir Kruchinin_, Feb 01 2012

%F Lim_{n->infinity} a(n)^(1/n)/n = (1+r)*(2+r)/exp(1) = 1.84542896220833..., where r = 0.794862961852611133... is the root of the equation (1+r)*(r+LambertW(-1,-r*exp(-r))) = -r. - _Vaclav Kotesovec_, Sep 24 2013

%e E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 188*x^4/4! + 3300*x^5/5! + ...

%e where A(x) = x - A(x)*log(1-A(x)).

%e The e.g.f. satisfies [from _Paul D. Hanna_, Jul 15 2012]:

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

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

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

%t Flatten[{0,Table[Sum[k!*Abs[StirlingS1[n-1,k]]*Binomial[n+k-1,n-1],{k,0,n-1}],{n,1,20}]}] (* _Vaclav Kotesovec_, Sep 24 2013 *)

%t (1+r)*(2+r)/E/.FindRoot[(1+r)*(r+LambertW[-1,-E^(-r)*r]) == -r, {r,1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n)/n, _Vaclav Kotesovec_, Sep 24 2013 *)

%o (PARI) a(n)=n!*polcoeff(serreverse(x+x*log(1-x +x*O(x^n))),n) \\ _Paul D. Hanna_, Aug 28 2008

%o (Maxima) a(n):=(sum(k!*(-1)^(n+k-1)*stirling1(n-1,k)*binomial(n+k-1,n-1),k,0,n-1)); /* _Vladimir Kruchinin_, Feb 01 2012 */

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*(-log(1-x+x*O(x^n)))^m/m!)); n!*polcoeff(A, n)} \\ _Paul D. Hanna_, Jul 15 2012

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*(-log(1-x+x*O(x^n)))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)} \\ _Paul D. Hanna_, Jul 15 2012

%Y Cf. A052802. - _Paul D. Hanna_, Aug 28 2008

%K easy,nonn

%O 0,3

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