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A123330 Expansion of eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3) in powers of q. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 = 0..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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, ...].

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).

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). - Michael Somos, Aug 11 2009

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. - Michael Somos, Aug 11 2009

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

a(4*n) = a(3*n) = a(n). a(12*n + 10) = a(6*n + 5) = 0. - Michael Somos, Aug 11 2009

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) = 123884(n). a(12*n + 7) = 4 * A121361(n). - Michael Somos, Aug 11 2009

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A294389 A152753 A113973 * A300821 A194564 A284690

Adjacent sequences:  A123327 A123328 A123329 * A123331 A123332 A123333

KEYWORD

nonn

AUTHOR

Michael Somos, Sep 26 2006

STATUS

approved

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Last modified July 21 04:40 EDT 2019. Contains 325189 sequences. (Running on oeis4.)