%I #12 Nov 01 2019 18:36:57

%S 1,1,2,7,26,123,622,3490,20468,125643,792606,5118050,33612998,

%T 223770400,1505528080,10213807498,69746716716,478693572991,

%U 3298184837434,22790090901504,157803590073220,1094189186549354,7593267782966708,52713912426435111,365948276764762712

%N G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^4)^2.

%C Compare definition of g.f. to:

%C (1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).

%C (2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

%C (3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 is the g.f. of the ternary tree numbers (A001764).

%C The first negative term is a(136). - _Georg Fischer_, Feb 16 2019

%H Paul D. Hanna, <a href="/A213094/b213094.txt">Table of n, a(n) for n = 0..300</a>

%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 123*x^5 + 622*x^6 + 3490*x^7 +...

%e Related expansions:

%e A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 237*x^4 + 1112*x^5 + 5614*x^6 +...

%e A(-x*A(x)^4)^2 = 1 - 2*x - 3*x^2 - 6*x^3 - 38*x^4 - 180*x^5 - 1095*x^6 -...

%t nmax = 24; sol = {a[0] -> 1};

%t Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x/A[-x A[x]^4]^2) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];

%t sol /. Rule -> Set;

%t a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 01 2019 *)

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^2,x,-x*subst(A^4,x,x+x*O(x^n))) );polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A000108, A001764, A213091, A213092, A213093, A213095, A213096.

%K sign

%O 0,3

%A _Paul D. Hanna_, Jun 05 2012