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%I #23 Mar 12 2021 22:24:48
%S 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,
%T -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,
%U 2,1,2,4,3,-3,0,2,1,-4,-5,2,-2,0,0,-4,-1,0,0,2,0
%N Expansion of eta(q) * eta(q^12) * eta(q^15) * eta(q^20) in powers of q.
%C 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).
%C 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
%H Seiichi Manyama, <a href="/A286128/b286128.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^(12 * k)) * (1 - x^(15 * k)) * (1 - x^(20 * k)).
%e 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
%t 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 *)
%o (PARI) 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
%o (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 */
%Y Cf. A030184 (eta(q) * eta(q^3) * eta(q^5) * eta(q^15)), A286129.
%K sign
%O 0,15
%A _Seiichi Manyama_, May 03 2017
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