A230204 Expansion of phi(-x) * f(x^3, x^5) in powers of x where phi(), f() are Ramanujan theta functions.

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

Offset: 0

Author

Michael Somos, Oct 11 2013

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - 2*x + x^3 + x^5 - 2*x^6 + 2*x^7 - 2*x^12 - x^14 - 2*x^15 + ...
G.f. = q - 2*q^17 + q^49 + q^81 - 2*q^97 + 2*q^113 - 2*q^193 - q^225 + ...
		

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

Crossrefs

Cf. A030204.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] QPochhammer[ -q^3, q^8] QPochhammer[ -q^5, q^8] QPochhammer[ q^8], {q, 0, n}];

PARI

{a(n) = local(m, j); if( n<0, 0, m = 16*n + 1; sum( k=0, sqrtint(m \ 4), if( issquare(m - 16*k^2, &j), if( k==0, 1, 2) * (-1)^k * ((j%8)==1 || (j%8==7)))))}

Formula

Euler transform of period 16 sequence [ -2, -1, -1, -1, -1, -2, -2, -2, -2, -2, -1, -1, -1, -1, -2, -2, ...].

a(n) = A030204(2*n).