A209939 Expansion of (f(x) * f(x^3))^3 in powers of q where f() is a Ramanujan theta function.

1, 3, 0, -2, 9, 0, -22, 0, 0, -26, -6, 0, 25, 27, 0, 46, 0, 0, 26, -66, 0, 22, 0, 0, -45, 0, 0, 0, -78, 0, 74, -18, 0, -122, 0, 0, -46, 75, 0, 142, 81, 0, 0, 0, 0, 44, 138, 0, 2, 0, 0, -194, 0, 0, -214, 78, 0, 0, -198, 0, 121, 0, 0, -146, 66, 0, 52, 0, 0, 22

Offset

0,2

Comments

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

Number 55 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^(-1/2) * ((eta(q^2) * eta(q^6))^3 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)))^3 in powers of q.

Euler transform of period 12 sequence [3, -6, 6, -3, 3, -12, 3, -3, 6, -6, 3, -6, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 48^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).

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

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

a(n) = (-1)^n * A030208(n). a(3*n + 2) = 0. a(3*n + 1) = 3 * a(n).

Conovlution cube of A208978. - Michael Somos, Jun 09 2015

Example

G.f. = 1 + 3*x - 2*x^3 + 9*x^4 - 22*x^6 - 26*x^9 - 6*x^10 + 25*x^12 + ...
G.f. = q + 3*q^3 - 2*q^7 + 9*q^9 - 22*q^13 - 26*q^19 - 6*q^21 + 25*q^25 + ...

Mathematica

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

Prog

(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)))^3, n))};

Crossrefs

Cf. A030208, A208978.

Sequence in context: A061980 A059683 A030208 * A231101 A193084 A126598

Adjacent sequences: A209936 A209937 A209938 * A209940 A209941 A209942

Keyword

sign

Author

Michael Somos, Mar 15 2012

Status

approved