A260516 Expansion of f(x, x^2) * f(x^2, x^10) in powers of x where f(,) is Ramanujan's general theta function.

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

Offset: 0

Author

Michael Somos, Jul 27 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 + 2*x^2 + x^3 + x^4 + x^5 + 2*x^7 + x^9 + x^10 + x^11 + ...
G.f. = q^17 + q^41 + 2*q^65 + q^89 + q^113 + q^137 + 2*q^185 + q^233 + ...
		

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

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 QPochhammer[ -x^2, x^4] / (QPochhammer[ x, x^2] QPochhammer[ x^12, x^24]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^4] EllipticTheta[ 2, Pi/4, x^3] / (x^(3/4) Sqrt[2] QPochhammer[ x]), {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A)^2 * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)), n))};

PARI

q='q+O('q^99); Vec(eta(q^3)^2*eta(q^4)^2*eta(q^24)/(eta(q)*eta(q^8)*eta(q^12))) \\ Altug Alkan, Aug 01 2018

Formula

Expansion of q^(-17/24) * eta(q^3)^2 * eta(q^4)^2 * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)) in powers of q.

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