OFFSET
1,2
FORMULA
E.g.f.: A(x) = Series_Reversion( exp(x) - exp(3*x^2/2) ).
a(n) ~ sqrt((3*s - 1)/(3 - 3*s + 9*s^2)) * n^(n-1) / (exp(s+1)-exp(3*s^2/2+1))^n, where s = 0.39177456704014800117... is the root of the equation 3*s*exp(3*s^2/2 - s) = 1. - Vaclav Kotesovec, Jan 12 2014
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 126*x^4/4! + 2049*x^5/5! +...
where A( exp(x) - exp(3*x^2/2) ) = x.
Related expansions:
exp(A(x)) = 1 + x + 3*x^2/2! + 18*x^3/3! + 195*x^4/4! + 3090*x^5/5! +...
exp(3*A(x)^2/2) = 1 + 3*x^2/2! + 18*x^3/3! + 195*x^4/4! + 3090*x^5/5! +...
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[E^x - E^(3*x^2/2), {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 12 2014 *)
PROG
(PARI) {a(n)=local(X=x+x*O(x^n)); if(n<1, 0, n!*polcoeff(serreverse(exp(X)-exp(3*X^2/2)), n))}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 07 2012
STATUS
approved