A262162 Expansion of f(-x^2)^5 * f(-x^12)^3 / (f(x)^2 * f(-x^8)^6) in powers of x where f() is a Ramanujan theta function.

1, -2, 0, 0, 0, 4, 0, 0, 1, -12, 0, 0, -3, 30, 0, 0, 4, -66, 0, 0, -3, 136, 0, 0, 5, -268, 0, 0, -12, 506, 0, 0, 14, -920, 0, 0, -10, 1622, 0, 0, 18, -2788, 0, 0, -37, 4688, 0, 0, 41, -7726, 0, 0, -34, 12506, 0, 0, 54, -19928, 0, 0, -98, 31306, 0, 0, 109

Offset: 0

Author

Michael Somos, Sep 13 2015

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 2*x + 4*x^5 + x^8 - 12*x^9 - 3*x^12 + 30*x^13 + 4*x^16 + ...
G.f. = q^-1 - 2*q^5 + 4*q^29 + q^47 - 12*q^53 - 3*q^71 + 30*q^77 + ...
		

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. A262150, A262151.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^5 QPochhammer[ x^12]^3 / (QPochhammer[ -x]^2 QPochhammer[ x^8]^6), {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^12 + A)^3 / (eta(x^2 + A) * eta(x^8 + A)^6), n))};

Formula

Expansion of q^(1/6) * eta(q)^2 * eta(q^4)^2 * eta(q^12)^3 / (eta(q^2) * eta(q^8)^6) in powers of q.

Euler transform of period 24 sequence [ -2, -1, -2, -3, -2, -1, -2, 3, -2, -1, -2, -6, -2, -1, -2, 3, -2, -1, -2, -3, -2, -1, -2, 0, ...].

a(4*n) = A262150(n). a(4*n + 1) = -2 * A262151(n). a(4*n + 2) = a(4*n + 3) = 0.