A134015 Expansion of (1 - phi(-q) * phi(q^4)) / 2 in powers of q where phi() is a Ramanujan theta function.

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

Offset: 1

Author

Michael Somos, Oct 02 2007

Keywords

• sign
• mult

Comments

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

Examples

			G.f. = x - 2*x^4 + 2*x^5 - 2*x^8 + x^9 + 2*x^13 - 2*x^16 + 2*x^17 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A002654, A008441, A113406, A134014.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^4]) / 2, {x, 0, n}]; (* Michael Somos, Oct 28 2015 *)
a[ n_] := If[ n < 1 || Mod[n, 4] > 1, 0, (Mod[n, 2] 3 - 2) DivisorSum[ n, KroneckerSymbol[ -4, #]&]]; (* Michael Somos, Oct 28 2015 *)

PARI

{a(n) = if( n<1 || n%4>1, 0, (n%2*3 - 2) * sumdiv(n, d, kronecker(-4, d)))};

PARI

{a(n) = -(-1)^n * if( n<1, 0, qfrep([1, 0; 0, 4], n)[n])};

Formula

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

a(n) is multiplicative with a(2) = 0, a(2^e) = -2 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).

a(4*n+2) = a(4*n+3) = 0.

G.f.: x / (1 + x^2) + x^3 / (1 + x^6) - 2 * x^4 / (1 + x^8) + ...

a(n) = -(-1)^n * A113406(n). -2 * a(n) = A134014(n) unless n=0. a(4*n) = -2 * A002654(n). a(4*n + 1) = A008441(n).