A256636 Expansion of phi(-x^3) / f(-x^2) in powers of x where phi(), f() are Ramanujan theta functions.

1, 0, 1, -2, 2, -2, 3, -4, 5, -6, 7, -10, 13, -14, 17, -22, 26, -30, 36, -44, 52, -60, 70, -84, 99, -112, 131, -156, 179, -204, 236, -274, 315, -358, 409, -472, 539, -608, 692, -792, 897, -1010, 1144, -1298, 1464, -1644, 1849, -2088, 2347, -2622, 2940, -3304

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(x, x^2) / psi(x) in powers of x where psi(), f() are Ramanujan theta functions.

Expansion of q^(1/12) * eta(q^3)^2 / (eta(q^2) * eta(q^6)) in powers of q.

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

G.f.: Product_{k>0} (1 + x^k + x^(2*k)) / ((1 + x^k) * (1 + x^(3*k))). [corrected by Vaclav Kotesovec, Jul 11 2016]

a(n) = (-1)^n * A258327(n). - Michael Somos, Jun 06 2015

Example

G.f. = 1 + x^2 - 2*x^3 + 2*x^4 - 2*x^5 + 3*x^6 - 4*x^7 + 5*x^8 - 6*x^9 + ...
G.f. = 1/q + q^23 - 2*q^35 + 2*q^47 - 2*q^59 + 3*q^71 - 4*q^83 + 5*q^95 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] / QPochhammer[ x^2], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 / (QPochhammer[ x^2] QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := SeriesCoefficient[ 2 x^(1/8) EllipticTheta[ 4, 0, x^3] / (EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ x, x^2]), {x, 0, n}]; (* Michael Somos, Jun 06 2015 *)

Prog

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

Crossrefs

Cf. A258327.

Sequence in context: A015744 A118301 A018121 * A258327 A102240 A026837

Adjacent sequences: A256633 A256634 A256635 * A256637 A256638 A256639

Keyword

sign

Author

Michael Somos, Apr 06 2015

Status

approved