OFFSET
0,2
COMMENTS
The 12th root of the power series E_6(x) has integral coefficients. See A109817. The 12th root of the g.f. of the present sequence also has integral coefficients. See A377223.
More generally if f(x) = g(x)^n, where g(x) = 1 + g_1*x + g_2*x^2 + ... is a power series with integral coefficients, then both the power series (1/x) * series_reversion(x*f(x)) and (1/x) * series_reversion(x/f(x)) are also equal to the n-th powers of integral power series.
EXAMPLE
The 12th root of the g.f. A(x)^(1/12) = (1 + 504*x + 524664*x^2 + 682155936*x^3 + 993260754360*x^4 + 1549502199011088*x^5 + 2532317522698504800*x^6 + ...)^(1/12) = 1 + 42*x + 34020*x^2 + 39770808*x^3 + 54603156174*x^4 + 82058923220904*x^5 + 130685055490645992*x^6 + ... lies in Z[[x]].
MAPLE
with(numtheory):
Order := 30:
E_6 := 1 - 504*add(sigma[5](n)*x^n, n = 1..30):
solve(series(x*E_6, x) = y, x):
seq(coeftayl(series((%/y), y), y = 0, n), n = 0..20);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Nov 08 2024
STATUS
approved