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%I #27 Sep 08 2022 08:45:31
%S 1,-2,-2,4,-2,8,4,-16,-2,-14,8,20,4,24,-16,-16,-2,-36,-14,36,8,32,20,
%T -48,4,-42,24,40,-16,56,-16,-64,-2,-40,-36,64,-14,72,36,-48,8,-84,32,
%U 84,20,56,-48,-96,4,-114,-42,72,24,104,40,-80,-16,-72,56,116,-16
%N Expansion of eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2 in powers of q.
%C Number 19 of the 74 eta-quotients listed in Table I of Martin (1996).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.67).
%H G. C. Greubel, <a href="/A131999/b131999.txt">Table of n, a(n) for n = 0..10000</a>
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%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) * phi(-q)^2 in powers of q where phi() is a Ramanujan theta function.
%F Euler transform of period 8 sequence [-2, -3, -2, -6, -2, -3, -2, -4, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^4 + u^2*v^2 + 2 * u^2*w^2 + 2 * u*v*w * (-u + 2*v - 2*w) - 2 * u*v^3.
%F a(n) = 2 * b(n) where b() is multiplicative with b(2^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 5 (mod 8).
%F a(2*n) = a(n) for all n in Z.
%F G.f.: 1 - 2* Sum_{k>0} k * x^k / (1 - x^k) * Kronecker(2, k).
%F G.f.: Product_{k>0} (1 - x^k)^4 * (1 + x^k)^2 * (1 + x^(2*k)) / (1 + x^(4*k))^2.
%F a(n) = -2 * A117000(n) unless n=0. a(n) = (-1)^n * A113416(n). a(2*n + 1) = - 2 * A113417(n).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(11/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A124340. - _Michael Somos_, Jun 28 2015
%F Convolution square is A259491. - _Michael Somos_, Jun 28 2015
%e G.f. = 1 - 2*q - 2*q^2 + 4*q^3 - 2*q^4 + 8*q^5 + 4*q^6 - 16*q^7 - 2*q^8 + ...
%t a[ n_] := If[ n < 1, Boole[n == 0], -2 DivisorSum[ n, # KroneckerSymbol[ 2, #] &]]; (* _Michael Somos_, Jun 28 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^3 / QPochhammer[ q^8]^2, {q, 0, n}]; (* _Michael Somos_, Jun 28 2015 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q]^2, {q, 0, n}]; (* _Michael Somos_, Jun 28 2015 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^4]^2, {q, 0, n}]; (* _Michael Somos_, Jun 28 2015 *)
%o (PARI) {a(n) = if( n<1, n==0, -2 * sumdiv(n, d, d * kronecker( 2, d)))};
%o (PARI) {a(n) = my(A, p, e); if( n<0, n==0, A = factor(n); -2 * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 1, abs(p%8-4)==3, (p^(e+1) - 1) / (p - 1), ((-p)^(e+1) - 1) / (-p - 1))))};
%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)^3 / eta(x^8 + A)^2, n))};
%o (Magma) A := Basis( ModularForms( Gamma1(8), 2), 61); A[1] - 2*A[2] - 2*A[3] + 4*A[4] - 2*A[5]; /* _Michael Somos_, Jun 28 2015 */
%Y Cf. A113416, A113417, A117000, A124340, A259491.
%K sign
%O 0,2
%A _Michael Somos_, Aug 06 2007
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