A133693 Expansion of (1 - phi(-q) * phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

1, -1, 2, -1, 0, -2, 0, -1, 3, 0, 2, -2, 0, 0, 0, -1, 2, -3, 2, 0, 0, -2, 0, -2, 1, 0, 4, 0, 0, 0, 0, -1, 4, -2, 0, -3, 0, -2, 0, 0, 2, 0, 2, -2, 0, 0, 0, -2, 1, -1, 4, 0, 0, -4, 0, 0, 4, 0, 2, 0, 0, 0, 0, -1, 0, -4, 2, -2, 0, 0, 0, -3, 2, 0, 2, -2, 0, 0, 0, 0

Offset: 1

Author

Michael Somos, Sep 20 2007

Keywords

• sign
• mult

Comments

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

For n nonzero, a(n) is nonzero if and only if n is in A002479.

Examples

			G.f. = q - q^2 + 2*q^3 - q^4 - 2*q^6 - q^8 + 3*q^9 + 2*q^11 - 2*q^12 - q^16 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A002325, A002479, A113411, A133692.

Programs

Mathematica

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Oct 30 2015 *)

PARI

{a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker( -2, d)))};

Formula

Expansion of (1 - eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2)) / 2 in powers of q.

Moebius transform is period 16 sequence [ 1, -2, 1, 0, -1, -2, -1, 0, 1, 2, 1, 0, -1, 2, -1, 0, ...].

a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8), a(p^e) = e + 1 if p == 1, 3 (mod 8).

a(8*n + 5) = a(8*n + 7) = 0. A133692(n) = -2 * a(n) unless n=0. a(n) = -(-1)^n * A002325(n). a(2*n + 1) = A113411(n).