A286129 Expansion of eta(q^3) * eta(q^4) * eta(q^5) * eta(q^60) in powers of q.

0, 0, 0, 1, 0, 0, -1, -1, -1, -1, 1, 0, 1, 0, 2, -1, 2, 2, 0, 0, -1, 1, -2, -1, 0, 0, -2, 0, -2, -2, 0, 2, -1, 0, -2, 1, 0, 0, 4, 2, 0, -2, 2, 1, 2, 1, 0, 1, -1, 2, 0, -4, 2, 0, -1, 0, 2, -4, 2, -2, 0, -2, -4, -3, -2, 0, 2, -1, 2, 5, -2, 2, -1, 0, -6, 1, -4, 0, 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 A286128. - 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^3 * Prod_{k>0} (1 - x^(3 * k)) * (1 - x^(4 * k)) * (1 - x^(5 * k)) * (1 - x^(60 * k)).

Example

G.f. = x^3 - x^6 - x^7 - x^8 - x^9 + x^10 + x^12 + 2*x^14 - x^15 + 2*x^16 + ... - Michael Somos, Oct 31 2019

Mathematica

With[{s = Flatten@ {#, Times @@ #} &@ Range[3, 5]}, Table[SeriesCoefficient[q QPochhammer[q^#1] QPochhammer[q^#2] QPochhammer[q^#3] QPochhammer[q^#4], {q, 0, n}], {n, -2, 76}] & @@ s] (* Michael De Vlieger, May 04 2017 *)
eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[eta[q^3]* eta[q^4]*eta[q^5]*eta[q^60], {q, 0, 50}], q] (* G. C. Greubel, Jul 29 2018 *)

Prog

(PARI) x='x + O('x^74); concat([0, 0, 0], Vec(eta(x^3)*eta(x^4) *eta(x^5)*eta(x^60))) \\ Indranil Ghosh, May 03 2017
({a(n) = my(A); n -= 3; if ( n < 0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A) * (eta(x^5 + A) * eta(x^60 + A), n))}; /* Michael Somos, Oct 31 2019 */

Crossrefs

Cf. A030184 (eta(q) * eta(q^3) * eta(q^5) * eta(q^15)), A286128.

Sequence in context: A082469 A206566 A088151 * A230322 A174950 A159906

Adjacent sequences: A286126 A286127 A286128 * A286130 A286131 A286132

Keyword

sign

Author

Seiichi Manyama, May 03 2017

Status

approved