A230205 Expansion of phi(-x) * f(x^1, x^7) in powers of x where phi(), f() are Ramanujan theta functions.

1, -1, -2, 0, 2, 2, 0, 1, -2, -2, -1, 0, 0, 0, 2, 0, 0, 2, 0, -2, 0, 0, 1, 0, 0, -2, 2, 1, -2, 0, 0, 0, -2, 0, 0, -2, 0, 2, 2, 0, 0, 0, 0, 4, 0, 1, 0, -2, 0, 0, -2, 0, -1, -2, -2, 0, 0, 0, 2, -2, 0, 0, 0, 2, 2, 2, 0, 0, 2, 0, -2, 0, 0, 0, 2, 0, -1, -4, 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 - x - 2*x^2 + 2*x^4 + 2*x^5 + x^7 - 2*x^8 - 2*x^9 - x^10 + ...
G.f. = q^9 - q^25 - 2*q^41 + 2*q^73 + 2*q^89 + q^121 - 2*q^137 - 2*q^153 + ...
		

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^1,q^8]* QPochhammer[-q^7,q^8]*QPochhammer[q^8], {q, 0, n}];

PARI

{a(n) = local(m, j); if( n<0, 0, m = 16*n + 9; sum( k=0, sqrtint(m \ 4), if( issquare(m - 16*k^2, &j), if( k==0, 1, 2) * (-1)^k * ((j%8)==3 || (j%8==5)))))}

Formula

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

a(n) = - A030204(2*n + 1).