OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..345
FORMULA
E.g.f. A(x) satisfies: A'(x) = A(x) + A(x)^5.
E.g.f. A(x) satisfies: A(x) = exp(x + Integral A(x)^4 dx).
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
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
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 152*x^3/3! + 2832*x^4/4! + 69152*x^5/5! +...
where A(x)^5 = 1 + 10*x + 140*x^2/2! + 2680*x^3/3! + 66320*x^4/4! +...
Also, A(x)^4 = 1 + 8*x + 96*x^2/2! + 1664*x^3/3! + 38400*x^4/4! +...
and log(A(x)) = 2*x + 8*x^2/2! + 96*x^3/3! + 1664*x^4/4! + 38400*x^5/5! +...
MATHEMATICA
CoefficientList[Series[E^x/(2-E^(4*x))^(1/4), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Dec 19 2013 *)
PROG
(PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); n!*polcoeff(exp(X)/(2-exp(4*X))^(1/4), n)}
for(n=0, 30, print1(a(n), ", "))
(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)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Dec 18 2013
STATUS
approved