A280339 Expansion of phi(x)^2 * chi(x^2)^4 * f(-x)^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.

1, 2, -1, -2, -5, -14, 4, 12, 5, 40, 0, -26, 11, -68, -15, 30, -18, 106, 3, -50, -10, -182, 29, 104, 10, 270, 11, -130, 37, -360, -51, 164, -16, 506, -30, -266, -65, -686, 62, 320, 53, 898, 22, -414, 50, -1206, -61, 612, -52, 1560, -4, -696, -81, -1958, 120

Offset: 0

Author

Michael Somos, Dec 31 2016

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 - x^2 - 2*x^3 - 5*x^4 - 14*x^5 + 4*x^6 + 12*x^7 + 5*x^8 + ...
G.f. = q^-1 + 2*q^3 - q^7 - 2*q^11 - 5*q^15 - 14*q^19 + 4*q^23 + 12*q^27 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A138515, A279955.

Programs

Mathematica

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

PARI

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

Formula

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

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

Euler transform of period 8 sequence [2, -4, 2, -8, 2, -4, 2, -4, ...].

a(n) = (-1)^n * A279955(n).

a(3*n + 1) / a(1) == A138515(n) (mod 3). a(3^3*n + 7) / a(7) == A138515(n) (mod 3^2).