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%I #44 Mar 12 2021 22:24:48
%S 1,-1,-2,0,1,4,0,0,-2,-4,2,0,0,-2,0,0,1,4,4,0,-4,0,0,0,0,-3,-4,0,0,4,
%T 0,0,-2,0,2,0,4,-2,0,0,2,4,0,0,0,-8,0,0,0,-1,-6,0,2,4,0,0,0,0,2,0,0,
%U -2,0,0,1,8,0,0,-4,0,0,0,4,-2,-4,0,0,0,0,0,-4
%N Expansion of f(-q) * f(-q^2) * chi(-q^3) in powers of q where chi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Denoted by a_6(n) in Kassel and Reutenauer 2015. - _Michael Somos_, Jun 04 2015
%H Seiichi Manyama, <a href="/A258210/b258210.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1505.07229v3">The zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1505.07229v3 [math.AG], 2015 , see page 31 7.2(d). [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1610.07793">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1610.07793 [math.NT], 2016, see p. 13 paragraph 3.3.4.
%H Christian Kassel, Christophe Reutenauer, <a href="http://arxiv.org/abs/1603.06357">The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z)</a>, arXiv:1603.06357 [math.NT], 2016.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of f(-q)^2 * f(-q^6) / f(-q, -q^5) in powers of q where f(,) is Ramanujan's general theta function.
%F Expansion of eta(q) * eta(q^2) * eta(q^3) / eta(q^6) in powers of q.
%F Euler transform of period 6 sequence [ -1, -2, -2, -2, -1, -2, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 12 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121444.
%F G.f.: Product_{k>0} (1 - x^k) * (1 - x^2*k) / (1 + x^(3*k)).
%F a(n) = (-1)^n * A258228(n). Convolution inverse of A077285.
%F a(4*n + 3) = 0. a(6*n + 2) = -2 * A122865(n). a(6*n + 4) = A122856(n). a(12*n + 1) = -1 * A002175(n).
%F a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = A104794(n). a(3*n + 1) = -A258277(n). a(3*n + 2) = -2*A258278(n). - _Michael Somos_, May 01 2016
%F G.f.: Product_{i>0} 1/(1 + Sum_{j>0} j*x^(j*i)). - _Seiichi Manyama_, Oct 08 2017
%e G.f. = 1 - q - 2*q^2 + q^4 + 4*q^5 - 2*q^8 - 4*q^9 + 2*q^10 - 2*q^13 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 / (QPochhammer[ q, q^6] QPochhammer[ q^5, q^6]), {q, 0, n}];
%t a[ n_] := SeriesCoefficient[ (1/2) EllipticThetaPrime[ 1, 0, q^(1/2)] / EllipticTheta[ 1, Pi/6, q^(1/2)], {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A) / eta(x^6 + A), n))};
%o (PARI) {a(n) = if( n<1, n==0, (-1)^n * (1 - (n%3==2)*3) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))}; /* _Michael Somos_, Jun 04 2015 */
%Y Cf. A002175, A077285, A104794, A121444, A258228, A258277, A258278, A290217.
%Y For the square of this series see A252650.
%K sign
%O 0,3
%A _Michael Somos_, May 23 2015
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