A254526 - OEIS

%I #12 Feb 16 2025 08:33:24

%S 1,0,0,0,0,12,0,-12,0,0,12,0,0,0,-12,12,0,12,0,-12,12,-12,0,0,0,12,0,

%T 0,-12,12,12,-12,0,0,12,0,0,0,-12,0,12,12,-12,-12,0,12,0,0,0,-12,12,

%U 12,0,12,0,0,-12,-12,12,0,12,0,-12,-12,0,24,0,-12,12,0

%N Fourier expansion of first basis element of space of weight 1 modular forms on Gamma1(12).

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A254526/b254526.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of 3 * (phi(q)^2 + phi(q^3)^2) / 2 -  (a(q) + a(q^2)) in powers of q where phi() is a Ramanujan theta function and a() is a cubic AGM theta function.

%F Moebius transform is period 12 sequence [ 0, 0, 0, 0, 12, 0, -12, 0, 0, 0, 0, 0, ...]. - _Michael Somos_, Jan 31 2015

%e G.f. = 1 + 12*q^5 - 12*q^7 + 12*q^10 - 12*q^14 + 12*q^15 + 12*q^17 - 12*q^19 + ...

%t a[ n_] := If[ n < 1, Boole[n==0], 12 Sum[ Boole[ Mod[d, 12] == 5] - Boole[ Mod[d, 12] == 7], {d, Divisors @ n}]];

%o (PARI) {a(n) = if( n<1, n==0, 12 * sumdiv( n, d, (d%12 == 5) - (d%12 == 7)))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^6 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^12 + A)) - 2 * eta(x^2 + A)^6 * eta(x^3 + A) / (eta(x + A)^3 * eta(x^6 + A)^2), n))};

%o (Magma) Basis( ModularForms( Gamma1(12), 1), 70)[1];

%K sign

%O 0,6

%A _Michael Somos_, Jan 31 2015