%I #6 Jun 05 2012 22:13:19

%S 1,1,3,10,41,76,-2183,-54998,-1045567,-15948296,-157645999,2035442014,

%T 217585291057,10000385378452,373813151971001,11759936127330346,

%U 269243105500780673,-519586631788126352,-649842878319124373855,-59793494397006229506890

%N E.g.f.: A(x) = exp( x/A(-x*A(x)) ).

%C Compare the e.g.f. to:

%C (1) W(x) = exp(x/W(-x*W(x)^2)^1) when W(x) = Sum_{n>=0} (1*n+1)^(n-1)*x^n/n!.

%C (2) W(x) = exp(x/W(-x*W(x)^4)^2) when W(x) = Sum_{n>=0} (2*n+1)^(n-1)*x^n/n!.

%C (3) W(x) = exp(x/W(-x*W(x)^6)^3) when W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 41*x^4/4! + 76*x^5/5! - 2183*x^6/6! +...

%e Related expansions:

%e 1/A(-x*A(x)) = 1 + x + x^2/2! + x^3/3! - 23*x^4/4! - 419*x^5/5! - 5159*x^6/6! +...

%e The logarithm of the e.g.f., log(A(x) = x/A(-x*A(x)), begins:

%e log(A(x)) = x + 2*x^2/2! + 3*x^3/3! + 4*x^4/4! - 115*x^5/5! - 2514*x^6/6! - 36113*x^7/7! +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(x/subst(A,x,-x*A+x*O(x^n))));n!*polcoeff(A,n)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A213109, A213110, A213111, A213112, A213113.

%K sign

%O 0,3

%A _Paul D. Hanna_, Jun 05 2012