A208978 Expansion of f(x) * f(x^3) where f() is a Ramanujan theta function.

1, 1, -1, 1, 1, -2, -1, -2, 0, 0, -1, 1, -1, 1, 0, -1, -1, 1, 2, 0, 1, -2, 1, 1, 0, 1, 2, 1, 0, 1, -1, 0, -1, 1, 0, 1, -2, -2, 2, 0, -1, -1, 0, -1, 0, 1, 2, -2, 1, 0, -2, -2, -1, 0, -1, 1, -1, -1, -1, 0, 0, 1, -1, 1, 0, 0, 2, 2, -1, 0, -1, -1, 2, -2, 0, 0, 2

Offset

0,6

Comments

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

Number 72 of the 74 eta-quotients listed in Table I of Martin (1996).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/6) * (eta(q^2) * eta(q^6))^3 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 1, -2, 2, -1, 1, -4, 1, -1, 2, -2, 1, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (432 t)) = 432^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

a(n) = b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6), b(p^e) = (-1)^(floor(p/6) * e) * t if p == 1 (mod 6) where t = e+1 if p = x^2 + 27*y^2, and t = [1, -1, 0] depending on e (mod 3) otherwise.

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

a(n) = (-1)^n * A030203(n).

Example

G.f. = 1 + x - x^2 + x^3 + x^4 - 2*x^5 - x^6 - 2*x^7 - x^10 + x^11 - x^12 + ...
G.f. = q + q^7 - q^13 + q^19 + q^25 - 2*q^31 - q^37 - 2*q^43 - q^61 + q^67 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^3] , {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)

Prog

(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 6*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%6==5, (1 + (-1)^e)/2, (-1)^(p\6*e) * if( (p-1) / znorder( Mod(2, p))%3, kronecker( e+1, 3), e+1))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))};

Crossrefs

Cf. A030203.

Sequence in context: A332998 A301652 A030203 * A333451 A101664 A091952

Adjacent sequences: A208975 A208976 A208977 * A208979 A208980 A208981

Keyword

sign

Author

Michael Somos, Mar 03 2012

Status

approved