OFFSET
0,2
COMMENTS
The cube root of F(x) = (1 + x)*(1 + 7*x)*(1 + 49*x) = (1 + 57*x + 399*x^2 + 343*x^3) is an integer series because F(x) == (1+x)^3 (mod 9).
In general, for k > 1, if g.f. = ((1 + x)*(1 + k*x)*(1 + k^2*x))^(1/3), then a(n) ~ (-1)^(n+1) * (k-1)^(2/3) * (k+1)^(1/3) * k^(2*n-1) / (3*Gamma(2/3)*n^(4/3)). - Vaclav Kotesovec, Feb 24 2024
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..406
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = (1 + x)*(1 + 7*x)*(1 + 49*x) = (1 + 57*x + 399*x^2 + 343*x^3).
(2) Product_{n>=1} A( (-7)^(n-1)*x^n )^3 = Sum_{n>=0} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2).
a(n) ~ (-1)^(n+1) * 2^(5/3) * 7^(2*n-1) / (3^(1/3) * Gamma(2/3) * n^(4/3)). - Vaclav Kotesovec, Feb 24 2024
EXAMPLE
G.f.: A(x) = 1 + 19*x - 228*x^2 + 6492*x^3 - 216372*x^4 + 7851180*x^5 - 300848772*x^6 + 11974587132*x^7 - 490113592788*x^8 + ...
where A(x)^3 = (1 + 57*x + 399*x^2 + 343*x^3).
RELATED SERIES.
We have the following infinite product involving the g.f. A(x)
A(x)^3 * A(-7*x^2)^3 * A(49*x^3)^3 * A(-343*x^4)^3 * A(2401*x^5)^3 * ... = 1 + 57*x - 19607*x^3 - 47079151*x^6 + 791260232049*x^10 + 93090977300134793*x^15 + ... + (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2) + ...
PROG
(PARI) {a(n) = polcoeff( ( (1 + x)*(1 + 7*x)*(1 + 49*x) +x*O(x^n))^(1/3), n)}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Feb 23 2024
STATUS
approved