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A209940 Expansion of psi(x^4) * phi(-x^4)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta function.

%I A209940 #28 Aug 05 2025 10:27:40
%S A209940 1,-2,4,-8,7,-10,12,-8,18,-18,16,-24,21,-20,28,-32,20,-32,36,-24,42,
%T A209940 -42,28,-48,57,-36,52,-40,36,-58,60,-56,48,-66,48,-72,74,-42,80,-80,
%U A209940 61,-82,72,-56,90,-96,64,-72,98,-70,100,-104,64,-106,108,-72,114,-96
%N A209940 Expansion of psi(x^4) * phi(-x^4)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta function.
%C A209940 Number 47 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A209940 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A209940 G. C. Greubel, <a href="/A209940/b209940.txt">Table of n, a(n) for n = 0..1000</a>
%H A209940 David Broadhurst and Daniele Dorigoni, <a href="https://arxiv.org/abs/2507.21352">Resurgent Lambert series with characters</a>, arXiv:2507.21352 [math.NT], 2025. See p. 44.
%H A209940 Yves 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 A209940 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A209940 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A209940 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A209940 Expansion of q^(-1/2) * eta(q)^2 * eta(q^4)^9 / (eta(q^2)^5 * eta(q^8)^2) in powers of q.
%F A209940 Euler transform of period 8 sequence [ -2, 3, -2, -6, -2, 3, -2, -4, ...].
%F A209940 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 512^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113419.
%F A209940 a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = (1 - (-3)^(e+1)) / 4, b(p^e) = (-1)^(e * [p/6]) * ((p*f)^(e+1) - 1) / (p*f - 1) where f = Kronecker( 18, p).
%F A209940 a(n) = (-1)^n * A258096(n) = (-1)^floor(n/2) * A113419(n) = (-1)^(n + floor(n/2)) * A113417(n).
%e A209940 G.f. = 1 - 2*x + 4*x^2 - 8*x^3 + 7*x^4 - 10*x^5 + 12*x^6 - 8*x^7 + 18*x^8 + ...
%e A209940 G.f. = q - 2*q^3 + 4*q^5 - 8*q^7 + 7*q^9 - 10*q^11 + 12*q^13 - 8*q^15 + 18*q^17 + ...
%t A209940 a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^4]^9 / (QPochhammer[ q^2]^5 QPochhammer[ q^8]^2), {q, 0, n}]; (* _Michael Somos_, May 19 2015 *)
%o A209940 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^9 / (eta(x^2 + A)^5 * eta(x^8 + A)^2), n))};
%o A209940 (PARI) {a(n) = my(A, p, e, f); if( n<0, 0, A = factor(2*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0^e, p==3, ((-p)^(e+1) - 1) / ((-p) - 1), p *= kronecker( 18, p); (-1)^(e*(p\6)) * (p^(e+1) - 1) / (p - 1))))};
%Y A209940 Cf. A113417, A113419, A258096.
%K A209940 sign
%O A209940 0,2
%A A209940 _Michael Somos_, Mar 16 2012