A286128 Expansion of eta(q) * eta(q^12) * eta(q^15) * eta(q^20) in powers of q.

0, 0, 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -2, 1, 1, -2, 1, 0, 0, -1, -2, 1, 1, 0, 0, 0, 2, 2, -1, -2, 0, 0, 4, -1, -1, 0, 0, -2, -1, 2, -2, -1, 2, -1, 0, -1, -2, -2, 1, 4, 0, 0, 0, 0, -2, 4, -4, 2, 1, 2, 4, 3, -3, 0, 2, 1, -4, -5, 2, -2, 0, 0, -4, -1, 0, 0, 2, 0

Offset

0,15

Comments

Early in 2005 Michael Somos discovered a remarkable eta-product identity: eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30) = eta(q) * eta(q^12) * eta(q^15) * eta(q^20) + eta(q^3) * eta(q^4) * eta(q^5) * eta(q^60).

G.f. is a period 1 Fourier series that satisfies f(-1 / (60 t)) = 60 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A286129. - Michael Somos, Oct 31 2019

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Michael Somos, A Remarkable eta-product Identity

Formula

G.f.: x^2 * Prod_{k>0} (1 - x^k) * (1 - x^(12 * k)) * (1 - x^(15 * k)) * (1 - x^(20 * k)).

Example

G.f. = x^2 - x^3 - x^4 + x^7 + x^9 - 2*x^14 + x^15 + x^16 - 2*x^17 + x^18 + ... - Michael Somos, Oct 31 2019

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[eta[q]* eta[q^12]*eta[q^15]*eta[q^20], {q, 0, 50}], q] (* G. C. Greubel, Jul 29 2018 *)

Prog

(PARI) my(q='q+O('q^50), A = eta(q)*eta(q^12)*eta(q^15)*eta(q^20)); concat([0, 0], Vec(A)) \\ G. C. Greubel, Jul 29 2018
(PARI) {a(n) = my(A); n -= 2; if ( n < 0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A) * eta(x^15 + A) * eta(x^20 + A), n))}; /* Michael Somos, Oct 31 2019 */

Crossrefs

Cf. A030184 (eta(q) * eta(q^3) * eta(q^5) * eta(q^15)), A286129.

Sequence in context: A176811 A057594 A259029 * A197547 A239723 A229541

Adjacent sequences: A286125 A286126 A286127 * A286129 A286130 A286131

Keyword

sign

Author

Seiichi Manyama, May 03 2017

Status

approved