A257536 Expansion of phi(-x^4)^2 * f(-x^1, -x^5) in powers of x where phi(), f() are Ramanujan theta functions.

1, -1, 0, 0, -4, 3, 0, 0, 5, 0, 0, 0, -4, -4, 0, 0, 9, -4, 0, 0, -12, 3, 0, 0, 8, 12, 0, 0, -8, -4, 0, 0, 8, -5, 0, 0, -12, 0, 0, 0, 13, 0, 0, 0, -8, -8, 0, 0, 16, -4, 0, 0, -12, 12, 0, 0, 13, 12, 0, 0, -20, -8, 0, 0, 8, -9, 0, 0, -16, 12, 0, 0, 16, 0, 0, 0

Offset: 0

Author

Michael Somos, Apr 28 2015

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - x - 4*x^4 + 3*x^5 + 5*x^8 - 4*x^12 - 4*x^13 + 9*x^16 - 4*x^17 + ...
G.f. = q - q^4 - 4*q^13 + 3*q^16 + 5*q^25 - 4*q^37 - 4*q^40 + 9*q^49 + ...
		

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

Crossrefs

Cf. A116597.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] EllipticTheta[ 4, 0, x^4]^2 EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}];

PARI

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

Formula

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

Euler transform of period 24 sequence [ -1, 0, 0, -4, -1, -1, -1, -2, 0, 0, -1, -5, -1, 0, 0, -2, -1, -1, -1, -4, 0, 0, -1, -3, ...].

a(4*n + 2) = a(4*n + 3) = 0. 2 * a(n) = A116597(3*n + 1).