A139032 Expansion of 1 + c(q^6) / c(q^3) where c() is a cubic AGM theta function.

1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, -1, 0, 0, 4, 0, 0, -4, 0, 0, -1, 0, 0, 8, 0, 0, -8, 0, 0, -2, 0, 0, 14, 0, 0, -14, 0, 0, -4, 0, 0, 24, 0, 0, -23, 0, 0, -6, 0, 0, 40, 0, 0, -38, 0, 0, -10, 0, 0, 63, 0, 0, -60, 0, 0, -16, 0, 0, 98, 0, 0, -92, 0, 0, -24, 0, 0, 150, 0, 0, -140, 0, 0, -36, 0, 0, 224, 0, 0, -208

Offset

0,11

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 1 + eta(q^3) * eta(q^18)^3 / (eta(q^6) * eta(q^9)^3) in powers of q.

Expansion of 1 + q * chi(-q^3) / chi(-q^9)^3 = psi(q) * chi(-q^3) / phi(-q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Expansion of eta(q^2)^2 * eta(q^3) * eta(q^18) / (eta(q) * eta(q^6) * eta(q^9)^2) in powers of q.

Euler transform of period 18 sequence [ 1, -1, 0, -1, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 0, -1, 1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (3/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A145977.

G.f.: Product_{k>0} P(2, x^k)^2 * P(18, x^k) / (P(3, x^k) * P(9, x^k)) where P(n, x) is n-th cyclotomic polynomial.

a(3*n) = 0 unless n = 0. a(3*n + 2) = 0. a(3*n + 1) = A092848(n). Convolution inverse of A145977.

Example

1 + q - q^4 + 2*q^10 - 2*q^13 - q^16 + 4*q^19 - 4*q^22 - q^25 + 8*q^28 + ...

Mathematica

eta[x_] := QPochhammer[x]; A139032[n_] := SeriesCoefficient[eta[q^2]^2 *eta[q^3]*eta[q^18]/(eta[q]*eta[q^6]*eta[q^9]^2), {q, 0, n}];
Table[A139032[n], {n, 0, 20}] (* G. C. Greubel, Aug 09 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A) / (eta(x + A) * eta(x^6 + A) * eta(x^9 + A)^2), n))}

Crossrefs

Cf. A092848, A145977.

Sequence in context: A108913 A237885 A341775 * A182035 A343493 A095808

Adjacent sequences: A139029 A139030 A139031 * A139033 A139034 A139035

Keyword

sign

Author

Michael Somos, Apr 07 2008

Status

approved