A262151 Expansion of f(-x^3)^3 / (f(x)^2 * f(-x^2)) in powers of x where f() is a Ramanujan theta function.

1, -2, 6, -15, 33, -68, 134, -253, 460, -811, 1394, -2344, 3863, -6253, 9964, -15653, 24269, -37178, 56331, -84489, 125529, -184867, 270027, -391391, 563205, -804925, 1142998, -1613195, 2263675, -3159023, 4385502, -6057865, 8328200, -11397371, 15529768

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 q^(-5/24) * eta(q)^2 * eta(q^3)^3 * eta(q^4)^2 / eta(q^2)^7 in power of q.

Euler transform of period 12 sequence [ -2, 5, -5, 3, -2, 2, -2, 3, -5, 5, -2, 0, ...].

a(n) ~ (-1)^n * exp(Pi*sqrt(3*n/2)) / (2^(11/4) * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Sep 23 2015

Example

G.f. = 1 - 2*x + 6*x^2 - 15*x^3 + 33*x^4 - 68*x^5 + 134*x^6 - 253*x^7 + ...
G.f. = q^5 - 2*q^29 + 6*q^53 - 15*q^77 + 33*q^101 - 68*q^125 + 134*q^149 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 / (QPochhammer[ -x]^2 QPochhammer[ x^2]), {x, 0, n}];
nmax = 40; CoefficientList[Series[Product[(1-x^(3*k))^3 * (1+x^(2*k))^2 / ((1-x^k)^3 * (1+x^k)^5), {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 + A)^2 * eta(x^3 + A)^3 * eta(x^4 + A)^2 / eta(x^2 + A)^7, n))};

Crossrefs

Sequence in context: A078406 A327598 A258348 * A246320 A101352 A192980

Adjacent sequences: A262148 A262149 A262150 * A262152 A262153 A262154

Keyword

sign

Author

Michael Somos, Sep 13 2015

Status

approved