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%I #16 Jan 29 2024 01:27:23
%S 1,-2,0,0,2,0,0,0,0,-4,4,0,0,-4,0,0,2,0,4,0,0,0,0,0,0,-6,0,0,0,0,0,0,
%T 0,0,4,0,4,-4,0,0,4,0,0,0,0,-8,0,0,0,-2,0,0,4,0,0,0,0,0,4,0,0,-4,0,0,
%U 2,0,0,0,0,0,0,0,4,-4,0,0,0,0,0,0,0,-4,4
%N Expansion of phi(-q) * phi(-q^9) in powers of q where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A258322/b258322.txt">Table of n, a(n) for n = 0..2500</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of eta(q)^2 * eta(q^9)^2 / (eta(q^2) * eta(q^18)) in powers of q.
%F Euler transform of period 18 sequence [ -2, -1, -2, -1, -2, -1, -2, -1, -4, -1, -2, -1, -2, -1, -2, -1, -2, -2, ...]. - _Michael Somos_, May 26 2015
%F a(n) = -2*b(n) where b() is multiplicative with a(0) = 1, b(2^e) = -1 if e>0, b(3^e) = 1 + (-1)^e if e>0, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4).
%F a(n) = -2 * A257900(n) unless n = 0 or n == 2 (mod 3).
%F a(3*n + 1) = -2 * A258277(n). a(3*n + 2) = 0. a(9*n) = -4 * A113652(n). a(9*n + 3) = a(9*n + 6) = 0.
%F Sum_{k=1..n} abs(a(k)) ~ (Pi/3) * n. - _Amiram Eldar_, Jan 29 2024
%e G.f. = 1 - 2*q + 2*q^4 - 4*q^9 + 4*q^10 - 4*q^13 + 2*q^16 + 4*q^18 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^9]^2 / (QPochhammer[ q^2] QPochhammer[ q^18]), {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^9 + A)^2 / (eta(x^2 + A) * eta(x^18 + A)), n))};
%o (PARI) {a(n) = my(A, p, e); if( n<1, n==0, if( n%3 == 2, 0, A = factor(n); -2 * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, -1, p%4==3, if(p==3, 2, 1)*(1-e%2), e+1))))};
%Y Cf. A113652, A257900, A258277.
%Y Cf. A000122, A000700, A010054, A121373.
%K sign
%O 0,2
%A _Michael Somos_, May 26 2015
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