A298421 Expansion of f(q, q^2) * chi(q^3)^3 * f(-q^4)^2 in powers of q where chi(), f(), f(,) are Ramanujan theta functions.

1, 1, 1, 3, 1, 2, 1, -2, -1, -5, 0, -4, -3, 2, -2, 4, -3, 2, -5, -4, 2, -12, -2, -4, 3, 5, 0, 9, 2, 6, 6, -6, -1, -4, 6, -4, -5, 6, -2, 18, 0, 6, 0, -4, 4, -10, -4, -4, 9, 7, 7, 8, 2, 6, 7, -4, -6, -18, 0, -8, -4, 6, -2, 10, -3, 8, -18, -8, 2, -8, -4, -12, 5

Offset

0,4

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(q^3) * f(-q^4)^2 * chi(-q^6)^2 / chi(-q) in powers of q where chi(), f(), are Ramanujan theta functions.

Expansion of psi(q) * psi(q^2) * phi(-q^6)^2 / psi(-q^3) in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of eta(q^2) * eta(q^4)^2 * eta(q^6)^5 / (eta(q) *eta(q^3) * eta(q^12)^3) in powers of q.

Euler transform of period 12 sequence [1, 0, 2, -2, 1, -4, 1, -2, 2, 0, 1, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 18432^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A298420.

Example

G.f. = 1 + q + q^2 + 3*q^3 + q^4 + 2*q^5 + q^6 - 2*q^7 - q^8 - 5*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3] QPochhammer[ q^4]^2 QPochhammer[ q^6, q^12]^2 QPochhammer[ -q, q], {q, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(-3/2) EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q] EllipticTheta[ 4, 0, q^6]^2 / EllipticTheta[ 2, Pi/4, q^(3/2)], {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)^3), n))};

Crossrefs

Cf. A298420.

Sequence in context: A230500 A010281 A353361 * A080131 A319956 A082882

Adjacent sequences: A298418 A298419 A298420 * A298422 A298423 A298424

Keyword

sign

Author

Michael Somos, Jan 18 2018

Status

approved