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A370147 Expansion of ( (1 + x)*(1 + 7*x)*(1 + 49*x) )^(1/3).

%I A370147 #19 Feb 25 2024 13:02:48
%S A370147 1,19,-228,6492,-216372,7851180,-300848772,11974587132,-490113592788,
%T A370147 20492868223308,-871404823013412,37562003034015900,
%U A370147 -1637401559515373172,72053378865932154348,-3196217668534369463748,142763786831538212246076,-6415201218873454789867284,289797678008730755585589900
%N A370147 Expansion of ( (1 + x)*(1 + 7*x)*(1 + 49*x) )^(1/3).
%C A370147 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).
%C A370147 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
%H A370147 Paul D. Hanna, <a href="/A370147/b370147.txt">Table of n, a(n) for n = 0..406</a>
%H A370147 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%F A370147 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A370147 (1) A(x)^3 = (1 + x)*(1 + 7*x)*(1 + 49*x) = (1 + 57*x + 399*x^2 + 343*x^3).
%F A370147 (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).
%F A370147 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
%e A370147 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 + ...
%e A370147 where A(x)^3 = (1 + 57*x + 399*x^2 + 343*x^3).
%e A370147 RELATED SERIES.
%e A370147 We have the following infinite product involving the g.f. A(x)
%e A370147 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) + ...
%o A370147 (PARI) {a(n) = polcoeff( ( (1 + x)*(1 + 7*x)*(1 + 49*x) +x*O(x^n))^(1/3), n)}
%o A370147 for(n=0, 40, print1(a(n), ", "))
%Y A370147 Cf. A370145, A370149, A370148.
%K A370147 sign
%O A370147 0,2
%A A370147 _Paul D. Hanna_, Feb 23 2024