%I #14 Feb 16 2025 08:33:08
%S 1,4,-4,-32,-4,104,32,-192,-4,292,-104,-480,32,680,192,-832,-4,1160,
%T -292,-1440,-104,1536,480,-2112,32,2604,-680,-2624,192,3368,832,-3840,
%U -4,3840,-1160,-4992,-292,5480,1440,-5440,-104,6728,-1536,-7392,480,7592,2112,-8832,32,9412,-2604,-9280
%N Expansion of (eta(q^2)^9 / (eta(q)^2 * eta(q^4)^4))^2 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A138504/b138504.txt">Table of n, a(n) for n = 0..10000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of (phi(q) * phi(-q^2)^2)^2 in powers of q where phi() is a Ramanujan theta function.
%F Euler transform of period 4 sequence [ 4, -14, 4, -6, ...].
%F a(n) = 4 * b(n) where a(0) = 1, b(n) is multiplicative with b(2^e) = -1 if e>0, b(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 4), b(p^e) = ((-p^2)^(e+1) - 1) / ( -p^2 - 1) if p == 3 (mod 4).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 32 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122854.
%F G.f.: 1 + 4 * Sum_{k>0} -(-1)^k * (2*k-1)^2 * x^(2*k-1) / (1 + x^(2*k-1)).
%F a(n) = (-1)^n * A120030(n). a(n) = 4 * A138505(n) unless n=0.
%e G.f. = 1 + 4*q - 4*q^2 - 32*q^3 - 4*q^4 + 104*q^5 + 32*q^6 - 192*q^7 - 4*q^8 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^9 / (QPochhammer[ q]^2 QPochhammer[ q^4]^4))^2, {q, 0, n}]; (* _Michael Somos_, May 24 2015 *)
%t a[ n_] := If[ n < 1, Boole[n == 0], -4 DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] (-1)^(n/#) &]]; (* _Michael Somos_, May 24 2015 *)
%o (PARI) {a(n) = if( n<1, n==0, -4 * sumdiv(n, d, d^2 * kronecker(-4, d) * (-1)^(n/d)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^4))^2, n))};
%Y Cf. A120030, A138505.
%K sign,changed
%O 0,2
%A _Michael Somos_, Mar 21 2008