login
A133079
Expansion of f(x)^3 - 3 * x * f(x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.
5
1, -5, -7, 0, 0, 11, 0, -13, 0, 0, 0, 0, 17, 0, 0, 19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, -29, 0, 0, 0, 0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, -37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 0, 0, 0
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
There is a plus sign on the left side and the first and third plus signs on the right side which should be minuses in Ramanujan's equation.
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 357, Entry 5, Eq. (5.1)
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(x) * a(-x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = -192 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A204850.
a(n) = b(24*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 8).
G.f.: Sum_{k in Z} Kronecker( 2, 2*k + 1) * (6*k + 1) * x^(k * (3*k + 1)/2).
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = -5 * a(n). a(n) = (-1)^n * A116916(n).
a(n) = A133089(3*n) = A204850(3*n). - Michael Somos, Jun 19 2015
EXAMPLE
G.f. = 1 - 5*x - 7*x^2 + 11*x^5 - 13*x^7 + 17*x^12 + 19*x^15 - 23*x^22 + ...
G.f. = q - 5*q^25 - 7*q^49 + 11*q^121 - 13*q^169 + 17*q^289 + 19*q^361 - ...
MATHEMATICA
a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, m (-1)^Boole[Mod[m, 8] > 4], 0]]; (* Michael Somos, Jun 19 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 - 3 x QPochhammer[ -x^9]^3, {x, 0, 3 n}]; (* Michael Somos, Jun 19 2015 *)
PROG
(PARI) {a(n) = if( issquare( 24*n + 1, &n), n * (-1) ^ (n%8 > 4), 0)};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 24*n + 1; A = factor(n); prod(k = 1, matsize(A) [1], [p, e] = A[k, ]; if( p < 5, 0, p *= kronecker( -2, p); if( e%2, 0, p^(e/2) ))))};
(PARI) {a(n) = my(A); if( n<0, 0, n *= 3; A = x * O(x^n); polcoeff( eta(-x + A)^3 - 3 * x * eta(-x^9 + A)^3, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 08 2007
STATUS
approved