A213266 Expansion of psi(q) * psi(q^9) / (psi(q^2) * chi(q^3) * psi(-q^9)) in powers of q where psi(), chi() are Ramanujan theta functions.

1, 1, -1, -1, 0, 1, 0, -1, 1, 2, 0, -3, 0, 2, 0, -3, 0, 5, 0, -4, -2, 4, 0, -5, 0, 7, 2, -7, 0, 5, 0, -10, 1, 12, 0, -10, 0, 14, -4, -17, 0, 21, 0, -22, 4, 24, 0, -34, 0, 33, 1, -36, 0, 45, 0, -45, -8, 52, 0, -55, 0, 62, 8, -71, 0, 70, 0, -88, 2, 96, 0, -98

Offset

0,10

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

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

a(n) = A182038(n) unless n=0. a(6*n) = 0 unless n=0. a(6*n + 4) = 0. a(6*n + 2) = -A092848(n).

Example

1 + q - q^2 - q^3 + q^5 - q^7 + q^8 + 2*q^9 - 3*q^11 + 2*q^13 - 3*q^15 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q^2]^3* eta[q^3]*eta[q^12]*eta[q^18]^3/(eta[q]*eta[q^4]^2*eta[q^6]^2*eta[q^9]^2*eta[q^36]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 19 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A) * eta(x^18 + A)^3 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^2 * eta(x^9 + A)^2 * eta(x^36 + A)), n))}
(PARI) q='q+O('q^99); Vec(eta(q^2)^3*eta(q^3)*eta(q^12)*eta(q^18)^3 /(eta(q)*eta(q^4)^2*eta(q^6)^2*eta(q^9)^2*eta(q^36))) \\ Altug Alkan, Mar 20 2018

Crossrefs

Cf. A092848, A182038.

Sequence in context: A092241 A336124 A256580 * A182038 A128144 A128145

Adjacent sequences: A213263 A213264 A213265 * A213267 A213268 A213269

Keyword

sign

Author

Michael Somos, Jun 07 2012

Status

approved