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%I #26 Mar 30 2017 14:29:30
%S 1,2,7,14,35,64,136,238,457,770,1377,2248,3822,6072,9920,15406,24386,
%T 37114,57240,85590,129152,190104,281542,408616,595425,853244,1225705,
%U 1736304,2462830,3452240,4841442,6721262,9329664,12837572,17653935,24092998,32850206
%N Expansion of (eta(q^5) * eta(q^10) / (eta(q) * eta(q^2)))^2 in powers of q.
%C Number 9 of the 14 eta-quotients listed in Table 2 of Moy 2013.
%H Seiichi Manyama, <a href="/A227213/b227213.txt">Table of n, a(n) for n = 1..10000</a>
%H Richard Moy, <a href="http://arxiv.org/abs/1309.4320">Congruences among power series coefficients of modular forms</a>, arXiv:1309.4320 [math.NT], 2013.
%F Euler transform of period 10 sequence [ 2, 4, 2, 4, 0, 4, 2, 4, 2, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 1/25 * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132041.
%F G.f.: x * (Product_{k>0} ( (1 - x^(5*k)) * (1 - x^(10*k))) / ((1 - x^k) * (1 - x^(2*k))))^2. [corrected by _Vaclav Kotesovec_, Sep 08 2015]
%F Convolution inverse of A132041.
%F a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(3/4) * 5^(9/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015
%e G.f. = q + 2*q^2 + 7*q^3 + 14*q^4 + 35*q^5 + 64*q^6 + 136*q^7 + 238*q^8 + ...
%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^5] QPochhammer[ q^10] / (QPochhammer[ q] QPochhammer[ q^2]))^2, {q, 0, n}]; (* _Michael Somos_, Jan 10 2015 *)
%t nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(5*k)) * (1 - x^(10*k)) / ((1 - x^k) * (1 - x^(2*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^5 + A) * eta(x^10 + A) / (eta(x + A) * eta(x^2 + A)))^2, n))};
%Y Cf. A132041.
%K nonn
%O 1,2
%A _Michael Somos_, Sep 19 2013
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