A255319 Expansion of psi(x^3) * f(x, x^5) in powers of x where psi(), f() are Ramanujan theta functions.

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

Offset: 0

Author

Michael Somos, Feb 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^3 + x^4 + x^5 + 2*x^8 + x^9 + x^10 + x^11 + x^14 + x^16 + ...
G.f. = q^17 + q^41 + q^89 + q^113 + q^137 + 2*q^209 + q^233 + q^257 + q^281 + ...
		

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^6] QPochhammer[ x^12]  QPochhammer[ -x, x^2], {x, 0, n}];

PARI

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

Formula

Expansion of chi(x) * f(-x^6) * f(-x^12) in powers of x where chi(), f() are Ramanujan theta functions.

Expansion of q^(-17/24) * eta(q^2)^2 * eta(q^6) * eta(q^12) / (eta(q) * eta(q^4)) in powers of q.

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