A258940 Expansion of f(-x^8) * f(-x^12) * f(-x^24) * f(-x^2, -x^6)^2 / (f(-x^2) * f(-x^3, -x^5) * f(-x^3, -x^21)) in powers of x where f() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

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

Offset: 0

Author

Michael Somos, Nov 07 2015

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - x^2 + 2*x^3 + x^4 - x^5 + x^6 + x^7 + x^9 + 2*x^12 - x^14 + ...
G.f. = q - q^5 + 2*q^7 + q^9 - q^11 + q^13 + q^15 + q^19 + 2*q^25 - q^29 + ...
		

Links

• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A128582.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ Product[(1 - x^k)^-{ 0, -1, 2, 1, 1, -1, 0, -1, 0, -1, 1, 0, 1, -1, 0, -1, 0, -1, 1, 1, 2, -1, 0, -2}[[Mod[k, 24, 1]]], {k, n}], {x, 0, n}];

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 2, 0, 1, -2, -1, -1, 1, 0, 1, 0, 1, -1, 0, -1, 1, 0, 1, 0, 1, -1, -1, -2, 1, 0][k%24 + 1]), n))};

Formula

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

a(3*n + 2) = - A128582(n).

a(12*n + 8) = a(12*n + 11) = 0.