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%I #38 Sep 08 2022 08:45:10
%S 1,-2,-2,4,2,0,-4,0,2,-6,0,4,4,0,0,0,2,-4,-6,4,0,0,-4,0,4,-2,0,8,0,0,
%T 0,0,2,-8,-4,0,6,0,-4,0,0,-4,0,4,4,0,0,0,4,-2,-2,8,0,0,-8,0,0,-8,0,4,
%U 0,0,0,0,2,0,-8,4,4,0,0,0,6,-4,0,4,4,0,0,0,0,-10,-4,4,0,0,-4,0,4,-4,0,0,0,0,0,0,4,-4,-2,12,2,0,-8,0
%N Expansion of eta(q)^2 * eta(q^2) / eta(q^4) in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C a(n) is nonzero if and only if n is in A002479. - _Michael Somos_, Dec 15 2011
%C Absolute values appear to give A033715 = 2*A002325.
%C Denoted by a_4(n) in Kassel and Reutenauer 2015. - _Michael Somos_, Jun 04 2015
%H G. C. Greubel, <a href="/A082564/b082564.txt">Table of n, a(n) for n = 0..1000</a>
%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(c). [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 3.3.3.
%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 phi(-q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Mar 30 2007
%F Euler transform of period 4 sequence [ -2, -3, -2, -2, ...]. - _Michael Somos_, Mar 30 2007
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(11/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033761. - _Michael Somos_, Aug 29 2014
%F G.f.: Product_{k>0} (1 - x^k)^2 / (1 + x^(2*k)). - _Michael Somos_, Mar 30 2007
%F a(n) = -2 * A129134(n) unless n=0. - _Michael Somos_, Mar 30 2007
%F a(n) = (-1)^floor( (n+1)/2 ) * A033715(n). - _Michael Somos_, Aug 29 2014
%F a(2*n) = A133692(n). a(2*n + 1) = -2 * A125095(n). - _Michael Somos_, Aug 29 2014
%F a(3*n + 1) = -2 * A258747(n). a(3*n + 2) = -2 * A258764(n). - _Michael Somos_, Jun 09 2015
%e G.f. = 1 - 2*q - 2*q^2 + 4*q^3 + 2*q^4 - 4*q^6 + 2*q^8 - 6*q^9 + 4*q^11 + 4*q^12 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4], {q, 0, n}]; (* _Michael Somos_, Aug 29 2014 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2], {q, 0, n}]; (* _Michael Somos_, Aug 29 2014 *)
%t a[ n_] := If[ n < 1, Boole[ n == 0], 2 (-1)^Quotient[ n + 1, 2] DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* _Michael Somos_, Aug 29 2014 *)
%o (PARI) {a(n) = if( n<1, n==0, 2 * (-1)^((n+1) \ 2) * sumdiv( n, d, kronecker( -2, d)))}; /* _Michael Somos_, Mar 30 2007 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) / eta(x^4 + A), n))};
%o (Magma) A := Basis( ModularForms( Gamma1(32), 1), 105); A[1] - 2*A[2] - 2*A[3] + 4*A[4] + 2*A[5] - 4*A[7] + 2*A[9] - 6*A[10] + 4*A[12] + 4*A[13] - 4*A[16]; /* _Michael Somos_, Aug 29 2014 */
%Y Cf. A002479, A033715, A033761, A125095, A129134, A133692, A258747, A258764.
%K sign
%O 0,2
%A _Benoit Cloitre_, May 05 2003
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