A245669 Expansion of q * f(q, q^5)^3 in powers of q where f() is Ramanujan's two-variable theta function.

1, 3, 3, 1, 0, 3, 6, 3, 3, 6, 6, 3, 0, 6, 6, 1, 6, 9, 6, 0, 0, 12, 12, 3, 7, 6, 9, 6, 0, 12, 6, 3, 6, 6, 12, 3, 0, 12, 12, 6, 6, 12, 18, 6, 0, 12, 12, 3, 7, 15, 12, 0, 0, 9, 12, 6, 12, 18, 6, 6, 0, 18, 18, 1, 12, 12, 18, 6, 0, 12, 12, 9, 12, 18, 15, 6, 0, 18

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * (chi(q) * psi(-q^3))^3 in powers of q where chi(), psi() are Ramanujan theta functions.

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

Euler transform of period 12 sequence [ 3, -3, 0, 0, 3, -3, 3, 0, 0, -3, 3, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 2^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A245668.

G.f.: x * (Sum_{k in Z} x^(3*k^2 + 2*k))^3.

Convolution cube of A089801.

a(2*n + 2) = A005885(n).

Example

G.f. = q + 3*q^2 + 3*q^3 + q^4 + 3*q^6 + 6*q^7 + 3*q^8 + 3*q^9 + 6*q^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ ((EllipticTheta[ 3, 0, q^(1/3)] - EllipticTheta[ 3, 0, q^3]) / 2)^3, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] EllipticTheta[ 2, Pi/4, q^(3/2)])^3 / (2^(3/2) q^(1/8)), {q, 0, n}];

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)))^3, n))};

Crossrefs

Cf. A005885, A089801, A245668.

Sequence in context: A343875 A343874 A114795 * A211648 A261634 A295848

Adjacent sequences: A245666 A245667 A245668 * A245670 A245671 A245672

Keyword

nonn

Author

Michael Somos, Jul 28 2014

Status

approved