A232356 Expansion of 2/9 * c(q) * c(q^2) - q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function and c(q) is a cubic AGM theta function.

1, 0, 5, -2, 6, 4, 8, -6, 17, 0, 12, 2, 14, 0, 30, -14, 18, 16, 20, -12, 40, 0, 24, -2, 31, 0, 53, -16, 30, 24, 32, -30, 60, 0, 48, 14, 38, 0, 70, -36, 42, 32, 44, -24, 102, 0, 48, -10, 57, 0, 90, -28, 54, 52, 72, -48, 100, 0, 60, 12, 62, 0, 136, -62, 84, 48

Offset: 1

Author

Michael Somos, Nov 22 2013

Keywords

• sign

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).

Examples

			G.f. = q + 5*q^3 - 2*q^4 + 6*q^5 + 4*q^6 + 8*q^7 - 6*q^8 + 17*q^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A008438, A033686, A098098, A111932, A121443, A123532, A144614, A229615, A232343.

Programs

Magma

Basis( ModularForms( Gamma0(6), 2), 70) [2];

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ d ( 2 Mod[ d, 2] Boole[Mod[ n/d, 3] > 0] - Mod[ n/d, 2] Boole[ Mod[d, 3] > 0]), {d, Divisors @n}]];
a[ n_] := SeriesCoefficient[ 2 q (QPochhammer[ q^3] QPochhammer[ q^6])^3 / (QPochhammer[ q] QPochhammer[ q^2]) - q (QPochhammer[ q^2] QPochhammer[ q^6])^4 / (QPochhammer[ q] QPochhammer[ q^3])^2, {q, 0, n}];

PARI

{a(n) = local(A); if( n<1, 0, n--; A=x*O(x^n); polcoeff( 2 * (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A)) - (eta(x^2 + A) * eta(x^6 + A))^4 / (eta(x + A) * eta(x^3 + A))^2, n))};

Sage

ModularForms( Gamma0(6), 2, prec=70).1;

Formula

a(n) = 2 * A121443(n) - A111932(n). a(2*n) = -2 * A229615(n). a(12*n + 2) = a(12*n + 10) = 0.

a(n) = A123532(n) + 7 * A229615(n). a(3*n + 2) = 6 * A232343(n-1). a(6*n + 5) = 6 * A098098(n). a(12*n + 4) = -2 * A144614(n). a(12*n + 6) = 4 * A008438(n). a(12*n + 8) = -6 * A033686(n). - Michael Somos, May 23 2014