A213265 Expansion of psi(q) * psi(q^2) * psi(q^6) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function.

1, 1, 1, -1, -3, -2, 3, 8, 5, -7, -18, -12, 15, 38, 24, -30, -75, -46, 57, 140, 86, -104, -252, -152, 183, 439, 262, -313, -744, -442, 522, 1232, 725, -852, -1998, -1168, 1365, 3182, 1852, -2150, -4986, -2886, 3336, 7700, 4436, -5106, -11736, -6736, 7719

Offset: 0

Author

Michael Somos, Jun 07 2012

Keywords

• sign

Comments

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

Examples

			1 + q + q^2 - q^3 - 3*q^4 - 2*q^5 + 3*q^6 + 8*q^7 + 5*q^8 - 7*q^9 + ...
		

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

Crossrefs

Cf. A187153.

Programs

Mathematica

a[n_]:= SeriesCoefficient[EllipticTheta[2, 0, Sqrt[q]]*EllipticTheta[2, 0, q]*EllipticTheta[2, 0, q^3]/(EllipticTheta[2, 0, q^(3/2)]^3), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 2017 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^4 + A)^2 * eta(x^12 + A)^2 / (eta(x + A) * eta(x^6 + A)^7), n))}

Formula

Expansion of 1 + c(q) * c(q^4)^2 / c(q^2)^3 in powers of q where c() is a cubic AGM theta function.

Expansion of eta(q^2) * eta(q^3)^3 * eta(q^4)^2 * eta(q^12)^2 / (eta(q) * eta(q^6)^7) in powers of q.

Euler transform of period 12 sequence [ 1, 0, -2, -2, 1, 4, 1, -2, -2, 0, 1, 0, ...].

a(n) = A187153(n) unless n=0.

Empirical: Sum_{n>=0} a(n)/exp(Pi*n) = 3/4 + (1/4)*sqrt(-9 + 6*sqrt(3)). - Simon Plouffe, Mar 02 2021