A182036 Expansion of q^(1/2) * eta(q^2)^2 * eta(q^6)^2 / (eta(q) * eta(q^9)^3) in powers of q.

1, 1, 0, 1, 0, 0, -1, -2, 0, 1, 4, 0, 0, -1, 0, -3, -8, 0, 4, 14, 0, 1, -4, 0, -6, -23, 0, 5, 40, 0, 1, -10, 0, -10, -60, 0, 11, 98, 0, 4, -24, 0, -19, -140, 0, 17, 224, 0, 4, -54, 0, -31, -304, 0, 31, 478, 0, 9, -112, 0, -50, -627, 0, 46, 968, 0, 11, -224

Offset

0,8

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

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(3*n) = A132179(n). a(3*n + 2) = 0.

Expansion of psi(x) * f(-x^6)^2 / f(-x^9)^3 in powers of x where psi(), f() are Ramanujan theta functions. - Michael Somos, Aug 10 2017

Example

G.f. = 1 + x + x^3 - x^6 - 2*x^7 + x^9 + 4*x^10 - x^13 - 3*x^15 - 8*x^16 + ...
G.f. = 1/q + q + q^5 - q^11 - 2*q^13 + q^17 + 4*q^19 - q^25 - 3*q^29 - 8*q^31 + ...

Mathematica

eta[x_] := x^(1/24)*QPochhammer[x]; A182036[n_] := SeriesCoefficient[q^(1/2)*(eta[q^2]* eta[q^6])^2/(eta[q]*eta[q^9]^3), {q, 0, n}]; Table[A182036[n], {n, 0, 50}] (* G. C. Greubel, Aug 09 2017 *)
a[ n_] := SeriesCoefficient[ 1/2 x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ x^6]^2 / QPochhammer[ x^9]^3, {x, 0, n}]; (* Michael Somos, Aug 10 2017 *)

Prog

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

Crossrefs

Cf. A132179.

Sequence in context: A018843 A152836 A152837 * A334173 A174996 A286815

Adjacent sequences: A182033 A182034 A182035 * A182037 A182038 A182039

Keyword

sign

Author

Michael Somos, Apr 07 2012

Status

approved