A260295 Expansion of f(-x^2)^3 * f(-x^6)^3 / f(-x)^2 in powers of x where f() is a Ramanujan theta function.

1, 2, 2, 4, 5, 6, 7, 6, 9, 8, 11, 14, 10, 14, 15, 16, 14, 14, 19, 20, 21, 22, 21, 20, 28, 26, 24, 22, 29, 30, 26, 32, 26, 38, 35, 36, 37, 28, 39, 40, 41, 42, 34, 40, 43, 42, 47, 42, 49, 50, 56, 44, 42, 54, 55, 62, 57, 46, 50, 60, 56, 62, 50, 70, 60, 56, 74, 54

Offset: 0

Author

Michael Somos, Oct 07 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 6*x^7 + ...
G.f. = q^11 + 2*q^23 + 2*q^35 + 4*q^47 + 5*q^59 + 6*q^71 + 7*q^83 + ...
		

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^2]^3 QPochhammer[ x^6]^3 / QPochhammer[ x]^2, {x, 0, n}];

PARI

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

PARI

q='q+O('q^99); Vec(eta(q^2)^3*eta(q^6)^3/eta(q)^2) \\ Altug Alkan, Jul 31 2018

Formula

Expansion of q^(-11/12) * eta(q^2)^3 * eta(q^6)^3 / eta(q)^2 in powers of q.

Euler transform of period 6 sequence [2, -1, 2, -1, 2, -4, ...]. - Michael Somos, Aug 01 2018