A261252 Expansion of f(-x^3) * f(-x^6) / (f(x) * f(-x^4)) in powers of x where f() is a Ramanujan theta function.

1, -1, 2, -4, 7, -10, 14, -22, 33, -45, 62, -88, 122, -163, 216, -290, 386, -502, 650, -846, 1093, -1393, 1768, -2248, 2844, -3565, 4454, -5566, 6927, -8566, 10562, -13014, 15986, -19543, 23832, -29032, 35272, -42700, 51578, -62226, 74906, -89909, 107712

Offset

0,3

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

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

Euler transform of period 6 sequence [ -1, 2, -2, 2, -1, 0, ...].

Given g.f. A(x), then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (3*u^2 - v)^3 * v^3 - 4 * u^2 * (v - u^2) * (2*u^2 - v).

a(n) = A261251(3*n).

Convolution inverse is A132179.

a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (6^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

Example

G.f. = 1 - x + 2*x^2 - 4*x^3 + 7*x^4 - 10*x^5 + 14*x^6 - 22*x^7 + ...
G.f. = q - q^7 + 2*q^13 - 4*q^19 + 7*q^25 - 10*q^31 + 14*q^37 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] QPochhammer[ x^6] / (QPochhammer[ -x] QPochhammer[ x^4]), {x, 0, n}];

Prog

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

Crossrefs

Cf. A132179, A261251.

Sequence in context: A288243 A170890 A079963 * A056750 A171926 A171930

Adjacent sequences: A261249 A261250 A261251 * A261253 A261254 A261255

Keyword

sign

Author

Michael Somos, Aug 12 2015

Status

approved