OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (f(x, x^2) * f(-x))^2 in powers of x where f() is the Ramanujan general theta function.
Expansion of q^(-1/6) * (eta(q^2) * eta(q^3)^2 / eta(q^6))^2 in powers of q.
Euler transform of period 6 sequence [ 0, -2, -4, -2, 0, -4, ...].
a(n) = A030188(3*n).
EXAMPLE
G.f. = 1 - 2*x^2 - 4*x^3 - x^4 + 8*x^5 + 6*x^6 + 4*x^7 - 7*x^8 - 8*x^9 + ...
G.f. = q - 2*q^13 - 4*q^19 - q^25 + 8*q^31 + 6*q^37 + 4*q^43 - 7*q^49 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] QPochhammer[ x^2])^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3]^2 / QPochhammer[ x^6])^2, {x, 0, n}];
PROG
(PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^2 / eta(x^6 + A))^2, n))};
(Magma) A := Basis( ModularForms( Gamma0(72), 2), 409); A[2] - 2*A[14];
(Magma) A := Basis( CuspForms( Gamma0(72), 2), 409); A[1];
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, May 18 2015
STATUS
approved