A123330 Expansion of eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3) in powers of q.

1, 2, 4, 2, 2, 0, 4, 4, 4, 2, 0, 0, 2, 4, 8, 0, 2, 0, 4, 4, 0, 4, 0, 0, 4, 2, 8, 2, 4, 0, 0, 4, 4, 0, 0, 0, 2, 4, 8, 4, 0, 0, 8, 4, 0, 0, 0, 0, 2, 6, 4, 0, 4, 0, 4, 0, 8, 4, 0, 0, 0, 4, 8, 4, 2, 0, 0, 4, 0, 0, 0, 0, 4, 4, 8, 2, 4, 0, 8, 4, 0, 2, 0, 0, 4, 0, 8, 0, 0, 0, 0, 8, 0, 4, 0, 0, 4, 4, 12, 0, 2, 0, 0, 4, 8

Offset: 0

Author

Michael Somos, Sep 26 2006

Keywords

• nonn
• easy

Comments

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

Examples

			G.f. = 1 + 2*q + 4*q^2 + 2*q^3 + 2*q^4 + 4*q^6 + 4*q^7 + 4*q^8 + 2*q^9 + ... - _Michael Somos_, Aug 11 2009
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A033687, A033762, A097195, A107760, A112604, A112605, A113973, A121361, A123331, A123884, A275486.

Cf. A000700, A000122, A010054, A121373.

Programs

Mathematica

QP = QPochhammer; s = QP[q^2]*(QP[q^3]^6/(QP[q]^2*QP[q^6]^3)) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

PARI

{a(n) = if( n<1, n==0, 2 * sumdiv(n, d, -(-1)^d * kronecker( -3, d)))}

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^6 / (eta(x + A)^2 * eta(x^6 + A)^3), n))}

Sage

A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] + 2*A[1] # Michael Somos, Sep 27 2013

Formula

Expansion of c(q)^2 / (3 * c(q^2)) in powers of q where c() is a cubic AGM theta function.

Expansion of phi(-x^3)^3 / phi(-x) where phi() is a Ramanujan theta function.

a(n) = 2*b(n) where b(n) is multiplicative and b(2^e) = (1 - 3*(-1)^e) / 2 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).

Euler transform of period 6 sequence [ 2, 1, -4, 1, 2, -2, ...].

Moebius transform is period 6 sequence [ 2, 2, 0, -2, -2, 0, ...].

a(n) = 2 * A123331(n) if n>0. (-1)^n * a(n) = A113973(n).

G.f.: Product_{k>0} (1 + x^k)/(1 - x^k) * ((1 - x^(3*k)) / (1 + x^(3*k)))^3.

G.f.: 1 + 2 * Sum_{k>0} x^k / (1 - x^k + x^(2*k)) = theta_3(-x^3)^3 / theta_3(-x).

From Michael Somos, Aug 11 2009: (Start)

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u - v)^2 - 2 * u * w * (v - w).

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A107760.

a(4*n) = a(3*n) = a(n). a(12*n + 10) = a(6*n + 5) = 0.

a(2*n + 1) = 2 * A033762(n). a(3*n + 1) = 2 * A033687(n). a(4*n + 1) = 2 * A112604(n). a(4*n + 3) = 2 * A112605(n). a(6*n + 1) = 2 * A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 4 * A121361(n). (End)

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - Amiram Eldar, Nov 14 2023