A283023 Expansion of f(-x, -x^5)^2 / (f(x^2, x^10) * f(x^6, x^18)) in powers of x where f(, ) is Ramanujan's general theta function.

1, -2, 0, 2, 0, -4, 1, 6, 0, -8, 0, 12, -1, -18, 0, 24, 0, -32, 0, 44, 0, -58, 0, 76, 1, -100, 0, 128, 0, -164, 0, 210, 0, -264, 0, 332, -1, -416, 0, 516, 0, -640, -1, 790, 0, -968, 0, 1184, 2, -1444, 0, 1752, 0, -2120, 1, 2560, 0, -3078, 0, 3692, -2, -4420, 0

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of chi(-x)^2 * chi(x^3)^2 * chi(-x^12) / chi(x^2) in powers of x where chi() is a Ramanujan theta function.

Expansion of phi(-x) * chi(x^3)^2 * chi(-x^12) / phi(-x^4) in powers of x where phi(), chi() are Ramanujan theta functions.

Expansion of phi(-x) * phi(x^3) / (phi(-x^4) * psi(-x^6)) ih powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of q^(3/4) * eta(q)^2 * eta(q^6)^4 * eta(q^8) / (eta(q^2) * eta(q^3)^2 * eta(q^4)^2 * eta(q^12) * eta(q^24)) in powers of q.

Euler transform of period 24 sequence [-2, -1, 0, 1, -2, -3, -2, 0, 0, -1, -2, 0, -2, -1, 0, 0, -2, -3, -2, 1, 0, -1, -2, 0, ...].

a(n) = A134178(2*n). a(6*n + 2) = A(6*n + 4) = 0.

a(2*n + 1) = -2 * A083365(n). a(4*n + 1) = -2 * A081055(n). a(4*n + 3) = 2 * A081056(n).

a(6*n) = A029838(n). a(12*n) = A258741(n). a(12*n + 6) = A259774(n). a(24*n + 12) = - A258939(n).

Example

G.f. = 1 - 2*x + 2*x^3 - 4*x^5 + x^6 + 6*x^7 - 8*x^9 + 12*x^11 + ...
G.f. = q^-3 - 2*q + 2*q^9 - 4*q^17 + q^21 + 6*q^25 - 8*q^33 + 12*q^41 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ -x^3, x^6]^2 QPochhammer[ x^12, x^24] / QPochhammer[ -x^2, x^4], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^3, x^6]^2 QPochhammer[ x^12, x^24] / EllipticTheta[ 4, 0, x^4], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(1/2) x^(3/4) EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^3] / (EllipticTheta[ 4, 0, x^4] EllipticTheta[ 2, Pi/4, x^3]), {x, 0, n}] // Simplify;

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^8 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^4 + A)^2 * eta(x^12 + A) * eta(x^24 + A)), n))};
(PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q)^2*eta(q^6)^4*eta(q^8)/(eta(q^2)*eta(q^3)^2*eta(q^4)^2*eta(q^12)*eta(q^24)))} \\ Altug Alkan, Mar 21 2018

Crossrefs

Cf. A029838, A081055, A081056, A083365, A134178, A258741, A258939, A259774.

Sequence in context: A364499 A238763 A029186 * A372554 A288182 A072680

Adjacent sequences: A283020 A283021 A283022 * A283024 A283025 A283026

Keyword

sign

Author

Michael Somos, Feb 26 2017

Status

approved