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A370148 Expansion of A(x) = [ Sum_{n>=0} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2) ]^(1/3).

%I A370148 #11 Feb 25 2024 13:06:27
%S A370148 1,19,-361,4896,-186048,6361181,-265706784,10569322565,-439680983904,
%T A370148 18480280546656,-790074277452000,34174424338394976,
%U A370148 -1494143747622128305,65898152303725266336,-2928713377590693411552,131019840536990930329051,-5895300394280706457304448,266614701826937350737301056
%N A370148 Expansion of A(x) = [ Sum_{n>=0} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2) ]^(1/3).
%H A370148 Paul D. Hanna, <a href="/A370148/b370148.txt">Table of n, a(n) for n = 0..406</a>
%F A370148 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A370148 (1) A(x)^3 = Sum_{n>=0} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2).
%F A370148 (2) A(x)^3 = Product_{n>=1} (1 + (-7)^(n-1)*x^n) * (1 - (-7)^n*x^n) * (1 + (-7)^(n+1)*x^n), by the Jacobi triple product identity.
%F A370148 (3) A(x) = Product_{n>=1} F( 2^(n-1)*x^n ), where F(x) = (1 + 57*x + 399*x^2 + 343*x^3)^(1/3) which is the g.f. of A370147.
%F A370148 a(n) ~ (-1)^(n+1) * c * 7^(2*n) / n^(4/3), where c = 0.2168488573077459727164856825904737112... - _Vaclav Kotesovec_, Feb 24 2024
%e A370148 G.f.: A(x) = 1 + 19*x - 361*x^2 + 4896*x^3 - 186048*x^4 + 6361181*x^5 - 265706784*x^6 + 10569322565*x^7 - 439680983904*x^8 + 18480280546656*x^9 + ...
%e A370148 The cube of g.f. A(x) equals the infinite product
%e A370148 A(x)^3 = (1 + x)*(1 + 7*x)*(1 + 7^2*x) * (1 - 7*x^2)*(1 - 7^2*x^2)*(1 - 7^3*x^2) * (1 + 7^2*x^3)*(1 + 7^3*x^3)*(1 + 7^4*x^3) * (1 - 7^3*x^4)*(1 - 7^4*x^4)*(1 - 7^5*x^4) * ...
%e A370148 Notice that the cube of A(x) yields the series
%e A370148 A(x)^3 = 1 + 57*x - 19607*x^3 - 47079151*x^6 + 791260232049*x^10 + 93090977300134793*x^15 - 76664422756665399911143*x^21 + ... + (-7)^(n*(n-1)/2)*(1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2) + ...
%o A370148 (PARI) {a(n) = my(A);
%o A370148 A = prod(m=1, n+1, (1 + (-7)^(m-1)*x^m) * (1 - (-7)^m*x^m) * (1 + (-7)^(m+1)*x^m) +x*O(x^n))^(1/3);
%o A370148 polcoeff(H=A, n)}
%o A370148 for(n=0, 30, print1(a(n), ", "))
%Y A370148 Cf. A370147, A370016, A370334, A370019, A370336.
%K A370148 sign
%O A370148 0,2
%A A370148 _Paul D. Hanna_, Feb 23 2024