A229893 Expansion of q^2 * f(-q) * f(-q^4) * f(-q^16) * f(-q^2, -q^14) in powers of q where f() is a Ramanujan theta function.

1, -1, -2, 1, 0, 2, 2, -1, -2, -1, 2, -1, -2, 0, 0, 0, 1, 3, 2, -2, 2, -6, -4, 3, 0, 4, 0, 3, 2, 0, -4, 0, -2, -2, 0, -3, 0, 2, 0, 0, 4, -5, -2, 1, 6, 0, 4, 0, -3, 2, 2, 5, -8, 2, 4, -6, 0, 3, -4, -9, -8, 0, 8, 0, -2, -5, 4, 6, 0, 10, -2, 4, 6, -3, -6, 2, -2

Offset: 2

Author

Michael Somos, Oct 02 2013

Keywords

• sign

Comments

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

Examples

			G.f. = q^2 - q^3 - 2*q^4 + q^5 + 2*q^7 + 2*q^8 - q^9 - 2*q^10 - q^11 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 2..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A108520, A229502.

Programs

Magma

Basis( CuspForms( Gamma1(16), 2), 79)[2];

Mathematica

a[ n_] := SeriesCoefficient[ q^2 QPochhammer[ q^2, q^16] QPochhammer[ q^14, q^16] QPochhammer[ q^16]^2 QPochhammer[ q] QPochhammer[ q^4], {q, 0, n}]

PARI

{a(n) = local(A, m); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^16 + A) * sum( k=0, n\2, if( issquare( 16*k + 9, &m), (-1)^k * x^(2*k), 0), A), n))}

Sage

CuspForms( Gamma1(16), 2, prec=79).1

Formula

Euler transform of period 16 sequence [-1, -2, -1, -2, -1, -1, -1, -2, -1, -1, -1, -2, -1, -2, -1, -4, ...].

a(2^n) = A108520(n-1). a(16*n + 1) = a(16*n + 15) = 0. -2 * a(n) = A229502(2*n). a(8*n) = 2 * A229502(n).