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Expansion of eta(q) * eta(q^2) * eta(q^15) * eta(q^30) in powers of q.
3

%I #25 Sep 08 2022 08:46:19

%S 0,0,1,-1,-2,1,0,2,1,0,0,-2,1,-2,-2,0,2,-2,1,4,-1,2,-2,0,0,0,0,-1,2,2,

%T -1,-4,-3,2,4,-2,-2,-6,0,0,0,8,-2,4,6,1,0,4,-3,-8,1,-4,2,-2,0,-2,-2,0,

%U -4,-2,2,4,4,2,-2,0,2,8,-6,0,2,-4,1,4,-4,-1,-4,0,2

%N Expansion of eta(q) * eta(q^2) * eta(q^15) * eta(q^30) in powers of q.

%H Seiichi Manyama, <a href="/A286137/b286137.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/retaprod.html">A Remarkable eta-product Identity</a>

%F G.f.: x^2 * Prod_{k>0} (1 - x^k) * (1 - x^(2 * k)) * (1 - x^(15 * k)) * (1 - x^(30 * k)).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 30 (t/i)^2 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Nov 26 2019

%F a(3*n) = -A030218(n). - _Michael Somos_, Mar 10 2020

%e G.f. = x^2 - x^3 - 2*x^4 + x^5 + 2*x^7 + x^8 - 2*x^11 + x^12 + ... - _Michael Somos_, Mar 10 2020

%p seq(coeff(series(x^2*mul((1-x^k)*(1-x^(2*k))*(1-x^(15*k))*(1-x^(30*k)),k=1..n), x,n+1),x,n),n=0..150); # _Muniru A Asiru_, Jul 29 2018

%t eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[eta[q] *eta[q^2]*eta[q^15]*eta[q^30], {q, 0, 50}], q] (* _G. C. Greubel_, Jul 29 2018 *)

%o (PARI) q='q+O('q^50); A = eta(q)*eta(q^2)*eta(q^15)*eta(q^30); concat([0,0], Vec(A)) \\ _G. C. Greubel_, Jul 29 2018

%o (Magma) A := Basis( CuspForms( Gamma0(30), 2), 80); A[2] - A[3]; /* _Michael Somos_, Nov 26 2019 */

%Y Cf. A030218, A122776.

%K sign

%O 0,5

%A _Seiichi Manyama_, May 03 2017