%I #13 Aug 09 2018 09:46:48

%S 1,1,4,63,1278,29764,758065,20611793,590579518,17707907024,

%T 553879330720,18066513887790,615744470668778,22014659625607877,

%U 830262409494773896,33243718957578687811,1422095813097928147636,65311403344808947050730,3227884786251446164710376

%N G.f.: A(x) = 1+x + Sum_{n>=2} (A(x)^n - 1)^n.

%H Vaclav Kotesovec, <a href="/A227619/b227619.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.9913753087... . - _Vaclav Kotesovec_, May 07 2014

%e G.f.: A(x) = 1 + x + 4*x^2 + 63*x^3 + 1278*x^4 + 29764*x^5 +...

%e where

%e A(x) = 1+x + (A(x)^2 - 1)^2 + (A(x)^3 - 1)^3 + (A(x)^4 - 1)^4 + (A(x)^5 - 1)^5 +...

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

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A122400.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 21 2013