A257398 Expansion of phi(-x^6)^2 / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.

1, 1, 1, 2, 2, 3, 0, 1, 2, 0, 2, 0, 3, 2, 2, 3, 0, 2, 2, 2, 0, 0, 1, 0, 2, 2, 1, 4, 2, 4, 0, 0, 2, 0, 4, 1, 0, 0, 4, 2, 1, 0, 2, 2, 0, 0, 0, 2, 2, 4, 2, 1, 2, 4, 2, 2, 0, 1, 0, 0, 4, 0, 2, 4, 0, 0, 0, 2, 0, 2, 3, 0, 0, 2, 2, 2, 2, 3, 2, 0, 4, 0, 4, 2, 2, 0, 0

Offset: 0

Author

Michael Somos, Apr 21 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + x^7 + 2*x^8 + 2*x^10 + ...
G.f. = q + q^25 + q^49 + 2*q^73 + 2*q^97 + 3*q^121 + q^169 + 2*q^193 + ...
		

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

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] EllipticTheta[ 4, 0, x^6] ^2, {x, 0, n}];

PARI

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

Formula

Expansion of phi(x^3) * f(x, x^2) in powers of x where phi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

Expansion of q^(-1/24) * eta(q^2) * eta(q^6)^4 / (eta(q) * eta(q^12)^2) in powers of q.

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