OFFSET
0,2
COMMENTS
Given g.f. A(x), note that A(x)^(1/3) is not an integer series.
FORMULA
G.f.: (1+2*x)^(2/3) / ((1-x)*(1-4*x))^(1/3).
G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^2 / 3 * x^n/n ).
a(n) ~ 3^(1/3) * 2^(2*n) / (n^(2/3) * Gamma(1/3)). - Vaclav Kotesovec, Oct 18 2020
EXAMPLE
G.f.: A(x) = 1 + 3*x + 6*x^2 + 18*x^3 + 57*x^4 + 195*x^5 + 684*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^2*x^3/3 + 5^2*x^4/4 + 11^2*x^5/5 + 21^2*x^6/6 + 43^2*x^7/7 +...+ Jacobsthal(n)^2*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].
MATHEMATICA
CoefficientList[Series[(1+2*x)^(2/3) / ((1-x)*(1-4*x))^(1/3), {x, 0, 30}], x] (* Vaclav Kotesovec, Oct 18 2020 *)
PROG
(PARI) {Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)), n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^2*x^k/k)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(((1+2*x)^2/((1-x)*(1-4*x) +x*O(x^n)))^(1/3), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 25 2012
STATUS
approved