OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.
FORMULA
Expansion of (2*a(q^2) - a(q))^2 = b(q)^4 / b(q^2)^2 in powers of q where a(), b() are cubic AGM theta functions.
Expansion of (eta(q)^6 * eta(q^6) / (eta(q^2)^3 * eta(q^3)^2))^2 in powers of q.
Euler transform of period 6 sequence [-12, -6, -8, -6, -12, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 432 (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229615.
G.f.: ( Product_{k>0} (1 + x^(3*k)) * (1 - x^k)^3 / ((1 + x^k)^3 * (1 - x^(3*k))))^2.
Convolution square of A122859.
EXAMPLE
G.f. = 1 - 12*q + 60*q^2 - 156*q^3 + 204*q^4 - 72*q^5 - 84*q^6 - 96*q^7 + ...
MATHEMATICA
a[ n_] := If[n < 1, Boole[ n == 0], -12 Sum[ {1, -7, 10, -7, 1, 2}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
a[ n_] := If[n < 1, Boole[ n == 0], -12 Sum[ {1, -3, 4, -3, 1, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^6 / EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}];
PROG
(PARI) {a(n) = if( n<1, n==0, -12 * sumdiv( n, d, n/d * [2, 1, -7, 10, -7, 1][d%6 + 1]))};
(PARI) {a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * [0, 1, -3, 4, -3, 1][d%6 + 1]))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^6 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A)^2))^2, n))};
(Sage) A = ModularForms( Gamma0(6), 2, prec=50).basis(); A[0] - 12*A[1] + 60*A[2];
(Magma) A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] - 12*A[2] + 60*A[3];
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 26 2013
STATUS
approved