%I #7 Oct 06 2020 09:07:39
%S 1,1,8,102,1712,34785,819384,21810124,645122272,20957720148,
%T 741260263600,28350052179438,1165931175542064,51320048879474206,
%U 2407857124657086480,119990501174741855400,6330579163195128292800,352584892981590315935084,20675941712941698695206368,1273517057922072215818491064,82210136955409063394289646720
%N G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^3)].
%H Vaclav Kotesovec, <a href="/A305603/b305603.txt">Table of n, a(n) for n = 0..378</a>
%F O.g.f. A(x) satisfies:
%F (1) [x^n] exp( n * Integral A(x)^3 dx ) * (n + 1 - A(x)) = 0 for n > 0.
%F (2) A(x) = 1 + x*A(x)^2*(A(x) + 3*x*A'(x))/(1 - x*A(x)^3)^2.
%F a(n) ~ c * 3^n * n^(4/3) * n!, where c = 0.1925904251831569484470022... - _Vaclav Kotesovec_, Oct 06 2020
%e G.f.: A(x) = 1 + x + 8*x^2 + 102*x^3 + 1712*x^4 + 34785*x^5 + 819384*x^6 + 21810124*x^7 + 645122272*x^8 + 20957720148*x^9 + 741260263600*x^10 + ...
%e such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^3)].
%e RELATED SERIES.
%e A(x)^3 = 1 + 3*x + 27*x^2 + 355*x^3 + 5964*x^4 + 120021*x^5 + 2790794*x^6 + 73301427*x^7 + 2141393220*x^8 + 68800518492*x^9 + ...
%e 1/(1 - x*A(x)^3) = 1 + x + 4*x^2 + 34*x^3 + 428*x^4 + 6957*x^5 + 136564*x^6 + 3115732*x^7 + 80640284*x^8 + 2328635572*x^9 + ...
%e A'(x)/A(x) = 1 + 15*x + 283*x^2 + 6343*x^3 + 162076*x^4 + 4614153*x^5 + 144287466*x^6 + 4908441479*x^7 + 180383821348*x^8 + 7122692545660*x^9 + ...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A^3+x*O(x^n)))); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^3) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A305602, A305604, A209881, A305110.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 05 2018