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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).
4
1, 19, -361, 4896, -186048, 6361181, -265706784, 10569322565, -439680983904, 18480280546656, -790074277452000, 34174424338394976, -1494143747622128305, 65898152303725266336, -2928713377590693411552, 131019840536990930329051, -5895300394280706457304448, 266614701826937350737301056
OFFSET
0,2
LINKS
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = Sum_{n>=0} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2).
(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.
(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.
a(n) ~ (-1)^(n+1) * c * 7^(2*n) / n^(4/3), where c = 0.2168488573077459727164856825904737112... - Vaclav Kotesovec, Feb 24 2024
EXAMPLE
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 + ...
The cube of g.f. A(x) equals the infinite product
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) * ...
Notice that the cube of A(x) yields the series
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) + ...
PROG
(PARI) {a(n) = my(A);
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);
polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Feb 23 2024
STATUS
approved