A182034 Expansion of c(q^2)^2 / (c(q) * c(q^3)) in powers of q where c() is a cubic AGM theta function.

1, -1, 1, 0, 1, -2, 0, 0, 1, 0, -2, 4, 0, 2, -8, 0, 1, 2, 0, -4, 14, 0, 4, -24, 0, 1, 6, 0, -8, 38, 0, 8, -63, 0, 2, 16, 0, -14, 92, 0, 14, -150, 0, 4, 36, 0, -24, 208, 0, 23, -329, 0, 6, 78, 0, -40, 440, 0, 38, -684, 0, 10, 160, 0, -63, 884, 0, 60, -1358, 0

Offset

0,6

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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(-q^3)^2 * psi(q^3)^4) / (psi(q) * f(-q^9)^3) in powers of q where psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, May 20 2015

Expansion of eta(q) * eta(q^6)^6 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^9)^3) in powers of q.

Euler transform of period 18 sequence [ -1, 1, 1, 1, -1, -3, -1, 1, 4, 1, -1, -3, -1, 1, 1, 1, -1, 0, ...].

a(n) = (-1)^n * A164615(n). a(3*n) = 0 unless n=0. a(3*n + 1) = - A092848(n). a(3*n + 2) = A216046(n).

Convolution inverse is A258100. - Michael Somos, May 20 2015

Example

G.f. = 1 - q + q^2 + q^4 - 2*q^5 + q^8 - 2*q^10 + 4*q^11 + 2*q^13 - 8*q^14 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (2 q^(1/8) QPochhammer[ q^6]^6) / (QPochhammer[ q^9]^3 EllipticTheta[ 2, 0, q^(1/2)] QPochhammer[ q^3]^2), {q, 0, n}]; (* Michael Somos, May 20 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3] EllipticTheta[ 2, 0, q^(3/2)]^3) / (4 q QPochhammer[ q^9]^3 EllipticTheta[ 2, 0, q^(1/2)]), {q, 0, n}]; (* Michael Somos, May 20 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^9 + A)^3), n))};

Crossrefs

Cf. A092848, A164615, A216046, A258100.

Sequence in context: A065293 A364377 A164615 * A171912 A306605 A363929

Adjacent sequences: A182031 A182032 A182033 * A182035 A182036 A182037

Keyword

sign

Author

Michael Somos, Apr 07 2012

Status

approved