%I #4 Feb 16 2019 08:24:30
%S 1,0,1,1,3,7,16,45,111,311,834,2329,6521,18429,52667,151095,437178,
%T 1270035,3710065,10882077,32044740,94700739,280749180,834793837,
%U 2488822697,7438604115,22283235185,66893731444,201208674387,606321286160,1830213820180,5533440540954,16754840359013,50803933761199,154251935227044,468929198610654,1427240650197467,4348833380280444,13265036911604648,40502401300634184
%N G.f. satisfies: A(x) = x + A( A(x)^3 + A(x)^4 ).
%C Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%C Compare to: F(x) = x + F( F(x)^3 - F(x)^9 ) holds when F(x) = x + F(x)^3 is a g.f. of the ternary tree numbers (A001764).
%H Paul D. Hanna, <a href="/A323692/b323692.txt">Table of n, a(n) for n = 1..730</a>
%F G.f. satisfies:
%F (1) A(x - A(x^3 + x^4)) = x.
%F (2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^3+x^4)^(n+1) / (n+1)!.
%F (3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^3+x^4)^(n+1)/x / (n+1)! ).
%F (4) A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + G(G(G(G(x)))) + ... where G(x) = A(x)^3 + A(x)^4.
%e G.f.: A(x) = x + x^3 + x^4 + 3*x^5 + 7*x^6 + 16*x^7 + 45*x^8 + 111*x^9 + 311*x^10 + 834*x^11 + 2329*x^12 + 6521*x^13 + 18429*x^14 + 52667*x^15 + ...
%e such that A(x) = x + A( A(x)^3 + A(x)^4 ).
%e RELATED SERIES.
%e A(x)^2 = x^2 + 2*x^4 + 2*x^5 + 7*x^6 + 16*x^7 + 39*x^8 + 110*x^9 + 277*x^10 + 786*x^11 + 2125*x^12 + 5996*x^13 + 16884*x^14 + 48044*x^15 + ...
%e A(x)^3 = x^3 + 3*x^5 + 3*x^6 + 12*x^7 + 27*x^8 + 70*x^9 + 198*x^10 + 510*x^11 + 1465*x^12 + 3999*x^13 + 11406*x^14 + 32328*x^15 + 92685*x^16 + ...
%e A(x)^4 = x^4 + 4*x^6 + 4*x^7 + 18*x^8 + 40*x^9 + 110*x^10 + 312*x^11 + 823*x^12 + 2392*x^13 + 6600*x^14 + 19032*x^15 + 54331*x^16 + ...
%e A(x)^3 + A(x)^4 = x^3 + x^4 + 3*x^5 + 7*x^6 + 16*x^7 + 45*x^8 + 110*x^9 + 308*x^10 + 822*x^11 + 2288*x^12 + 6391*x^13 + 18006*x^14 + 51360*x^15 + ...
%e A(x^3 + x^4) = x^3 + x^4 + x^9 + 3*x^10 + 3*x^11 + 2*x^12 + 4*x^13 + 6*x^14 + 7*x^15 + 16*x^16 + 30*x^17 + 37*x^18 + 57*x^19 + 108*x^20 + ...
%e where Series_Reversion(A(x)) = x - A(x^3 + x^4).
%o (PARI) {a(n) = my(A=x); for(i=1, n, A = x + subst(A, x, A^3 + A^4 +x*O(x^n))); polcoeff(H=A, n)}
%o for(n=1, 40, print1(a(n), ", "))
%Y Cf. A155134, A271844,
%K nonn
%O 1,5
%A _Paul D. Hanna_, Feb 16 2019