Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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