A133691 Expansion of (1 - (phi(-q) * phi(q^2))^2) / 4 in powers of q where phi() is a Ramanujan theta function.

1, -2, 4, -6, 6, -8, 8, -6, 13, -12, 12, -24, 14, -16, 24, -6, 18, -26, 20, -36, 32, -24, 24, -24, 31, -28, 40, -48, 30, -48, 32, -6, 48, -36, 48, -78, 38, -40, 56, -36, 42, -64, 44, -72, 78, -48, 48, -24, 57, -62, 72, -84, 54, -80, 72, -48, 80, -60, 60, -144

Offset

1,2

Comments

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

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.

Formula

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

a(n) is multiplicative with a(2) = -2, a(2^e) = -6 if e>1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.

a(n) = -4 * A133690(n) = -(-1)^n * A111973(n). a(2*n) = -2 * A046897(n). a(2*n + 1) = A008438(n). a(4*n) = -6 * A000593(n). a(4*n + 1) = A112610(n). a(4*n + 3) = 4 * A097723(n).

Dirichlet g.f.: zeta(s) * zeta(s-1) * (1 - 5/2^s + 1/2^(2*s-1) + 1/2^(3*s-3)). - Amiram Eldar, Oct 28 2023

Example

G.f. = q - 2*q^2 + 4*q^3 - 6*q^4 + 6*q^5 - 8*q^6 + 8*q^7 - 6*q^8 + 13*q^9 - ...

Mathematica

a[ n_] := Which[ n < 1, 0, OddQ[n], DivisorSigma[ 1, n], True, -2 DivisorSum[ n/2, # Boole[Mod[#, 4] > 0] &]]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := SeriesCoefficient[ (1 - (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2) / 4, {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, n%2, sigma(n), -2 * sumdiv(n/2, d, if(d%4, d)))};

Crossrefs

Cf. A000593, A008438, A046897, A097723, A111973, A112610, A133690.

Sequence in context: A339743 A131450 A114218 * A111973 A349787 A161655

Adjacent sequences: A133688 A133689 A133690 * A133692 A133693 A133694

Keyword

sign,easy,mult

Author

Michael Somos, Sep 20 2007

Status

approved