A182032 Expansion of eta(q^2)^3 * eta(q^9) * eta(q^12)^4 / (eta(q) * eta(q^4)^2 * eta(q^6)^2 * eta(q^18) * eta(q^36)^2) in powers of q.

1, 1, -1, 0, 1, 0, 1, 1, 0, 0, -1, 0, 1, -1, 0, 0, 0, 0, -1, 1, 0, 0, 2, 0, -1, 0, 0, 0, -2, 0, 0, -3, 0, 0, -1, 0, 1, 4, 0, 0, 4, 0, 2, 1, 0, 0, -4, 0, 0, -6, 0, 0, -1, 0, -2, 5, 0, 0, 8, 0, -3, 1, 0, 0, -8, 0, -1, -10, 0, 0, -2, 0, 4, 11, 0, 0, 14, 0, 4, 4, 0, 0, -14, 0, 1, -19, 0, 0, -4, 0, -4, 17, 0, 0, 24, 0, -6, 4, 0, 0, -23, 0

Offset

-2,23

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 = -2..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-2) * chi(q) * chi(-q^2) * psi(q^6)^2 / (psi(q^9) * psi(q^18)) in powers of q where psi(), chi() are Ramanujan theta functions.

Euler transform of period 36 sequence [ 1, -2, 1, 0, 1, 0, 1, 0, 0, -2, 1, -2, 1, -2, 1, 0, 1, 0, 1, 0, 1, -2, 1, -2, 1, -2, 0, 0, 1, 0, 1, 0, 1, -2, 1, 0, ...].

a(6*n) = 0 unless n=0. a(6*n + 1) = a(6*n + 3) = 0.

A062242(n) = a(3*n - 1). A132179(n) = a(6*n - 1). A062242(n) = a(6*n - 2). A092848(n) = a(6*n + 2).

Example

q^-2 + q^-1 - 1 + q^2 + q^4 + q^5 - q^8 + q^10 - q^11 - q^16 + q^17 + ...

Mathematica

QP = QPochhammer; s = QP[q^2]^3*QP[q^9]*(QP[q^12]^4 / (QP[q]*QP[q^4]^2* QP[q^6]^2*QP[q^18]*QP[q^36]^2)) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<-2, 0, n+=2; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^9 + A) * eta(x^12 + A)^4 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^2 * eta(x^18 + A) * eta(x^36 + A)^2), n))}

Crossrefs

Cf. A062242, A092848, A132179.

Sequence in context: A091393 A359735 A284557 * A265245 A110270 A284825

Adjacent sequences: A182029 A182030 A182031 * A182033 A182034 A182035

Keyword

sign

Author

Michael Somos, Apr 07 2012

Status

approved