%I #22 Sep 01 2016 11:00:42
%S 1,2,12,152,2832,69152,2089152,75204992,3142025472,149428961792,
%T 7969790856192,471098477484032,30567292903821312,2159857294035525632,
%U 165083372031671058432,13570774387950150582272,1193933787763434969956352,111932230270819401046556672
%N E.g.f.: exp(x) / (2 - exp(4*x))^(1/4).
%H G. C. Greubel, <a href="/A229558/b229558.txt">Table of n, a(n) for n = 0..345</a>
%F E.g.f. A(x) satisfies: A'(x) = A(x) + A(x)^5.
%F E.g.f. A(x) satisfies: A(x) = exp(x + Integral A(x)^4 dx).
%F a(n) ~ GAMMA(3/4) * 4^n * n^(n-1/4) / (sqrt(Pi) * exp(n) * log(2)^(n+1/4)). - _Vaclav Kotesovec_, Dec 19 2013
%F a(n) = 1/2^(1/4) * Sum_{k >= 0} (1/32)^k*A034385(k)*(4*k + 1)^n = 1/2^(1/4)*Sum_{k >= 0} (-1/2)^k*binomial(-1/4, k)*(4*k + 1)^n. Cf. A124212 and A124214. - _Peter Bala_, Aug 30 2016
%e E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 152*x^3/3! + 2832*x^4/4! + 69152*x^5/5! +...
%e where A(x)^5 = 1 + 10*x + 140*x^2/2! + 2680*x^3/3! + 66320*x^4/4! +...
%e Also, A(x)^4 = 1 + 8*x + 96*x^2/2! + 1664*x^3/3! + 38400*x^4/4! +...
%e and log(A(x)) = 2*x + 8*x^2/2! + 96*x^3/3! + 1664*x^4/4! + 38400*x^5/5! +...
%t CoefficientList[Series[E^x/(2-E^(4*x))^(1/4), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Dec 19 2013 *)
%o (PARI) {a(n)=local(A=1+x,X=x+x*O(x^n));n!*polcoeff(exp(X)/(2-exp(4*X))^(1/4),n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A+A^5+x*O(x^n))); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A124214, A034385, A124212.
%K nonn,easy
%O 0,2
%A _Paul D. Hanna_, Dec 18 2013