%I #5 Jun 06 2012 01:04:21

%S 1,1,5,73,1497,48321,2016733,106687113,6745180529,495988880833,

%T 41495596689141,3880618840698249,400537444634948041,

%U 45126092520882513921,5501154522933362385485,720279890636684703825481,100658531630809161730405857,14934726665907895887483076737

%N E.g.f.: A(x) = exp( x/A(-x*A(x)^6)^2 ).

%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 + 5*x^2/2! + 73*x^3/3! + 1497*x^4/4! + 48321*x^5/5! +...

%e Related expansions:

%e A(x)^2 = 1 + 2*x + 12*x^2/2! + 176*x^3/3! + 3728*x^4/4! + 118912*x^5/5! +...

%e A(x)^6 = 1 + 6*x + 60*x^2/2! + 1008*x^3/3! + 23952*x^4/4! + 775296*x^5/5! +...

%e 1/A(-x*A(x)^6)^2 = 1 + 2*x + 20*x^2/2! + 296*x^3/3! + 7824*x^4/4! +...

%e The logarithm of the e.g.f., log(A(x)) = x/A(-x*A(x)^6)^2, begins:

%e log(A(x)) = x + 4*x^2/2! + 60*x^3/3! + 1184*x^4/4! + 39120*x^5/5! + 1639872*x^6/6! +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(x/subst(A^2,x,-x*A^6+x*O(x^n))));n!*polcoeff(A,n)}

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

%Y Cf. A213108, A213109, A213110, A213112, A213113.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 05 2012