A262150 Expansion of f(-x^3)^3 / (f(-x^2) * f(-x^4)^2) in powers of x where f() is a Ramanujan theta function.

1, 0, 1, -3, 4, -3, 5, -12, 14, -10, 18, -37, 41, -34, 54, -98, 109, -92, 138, -237, 260, -230, 329, -531, 583, -526, 728, -1129, 1233, -1143, 1537, -2292, 2503, -2355, 3097, -4486, 4889, -4677, 6031, -8502, 9263, -8962, 11372, -15680, 17066, -16703, 20893

Offset

0,4

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(1/24) * eta(q^3)^3 / (eta(q^2) * eta(q^4)^2) in powers of q.

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

a(n) = A143066(3*n).

a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(3/4) * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Sep 13 2015

Example

G.f. = 1 + x^2 - 3*x^3 + 4*x^4 - 3*x^5 + 5*x^6 - 12*x^7 + 14*x^8 + ...
G.f. = q^-1 + q^47 - 3*q^71 + 4*q^95 - 3*q^119 + 5*q^143 - 12*q^167 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 / (QPochhammer[ x^2] QPochhammer[ x^4]^2), {x, 0, n}];
nmax = 60; CoefficientList[Series[Product[(1-x^(3*k))^3 / ((1-x^(2*k))^3 * (1+x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 13 2015 *)

Prog

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

Crossrefs

Cf. A143066.

Sequence in context: A226798 A045997 A360059 * A325594 A104076 A238161

Adjacent sequences: A262147 A262148 A262149 * A262151 A262152 A262153

Keyword

sign

Author

Michael Somos, Sep 13 2015

Status

approved