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A257900 Expansion of 1/2 - (phi(-q)^2 + phi(-q^9)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

%I A257900 #31 Feb 16 2025 08:33:25
%S A257900 1,-1,0,-1,2,0,0,-1,2,-2,0,0,2,0,0,-1,2,-2,0,-2,0,0,0,0,3,-2,0,0,2,0,
%T A257900 0,-1,0,-2,0,-2,2,0,0,-2,2,0,0,0,4,0,0,0,1,-3,0,-2,2,0,0,0,0,-2,0,0,2,
%U A257900 0,0,-1,4,0,0,-2,0,0,0,-2,2,-2,0,0,0
%N A257900 Expansion of 1/2 - (phi(-q)^2 + phi(-q^9)^2) / 4 in powers of q where phi() is a Ramanujan theta function.
%C A257900 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A257900 G. C. Greubel, <a href="/A257900/b257900.txt">Table of n, a(n) for n = 1..2500</a>
%H A257900 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A257900 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A257900 Expansion of 1/2 - (eta(q)^4 * eta(q^18)^2 + eta(q^2)^2 * eta(q^9)^4) / (2 * eta(q^2) * eta(q^18))^2 in powers of q. - _Michael Somos_, Jun 02 2015
%F A257900 a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 1 + (-1)^e if e>0, a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4).
%F A257900 a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(9*n) = 2 * A113652(n). a(9*n + 3) = a(9*n + 6) = 0.
%F A257900 -2 * a(n) = A258322(n) unless n = 0 or n == 2 (mod 3).
%F A257900 Sum_{k=1..n} abs(a(k)) ~  (5*Pi/18) * n. - _Amiram Eldar_, Jan 29 2024
%e A257900 G.f. = q - q^2 - q^4 + 2*q^5 - q^8 + 2*q^9 - 2*q^10 + 2*q^13 - q^16 + ...
%t A257900 a[ n_] := SeriesCoefficient[ 1/2 - (EllipticTheta[ 4, 0, q]^2 + EllipticTheta[ 4, 0, q^9]^2) / 4, {q, 0, n}]; (* _Michael Somos_, Jun 02 2015 *)
%o A257900 (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, -1, p%4==3, if( p>3, 1, 2) * (1-e%2), e+1)))};
%Y A257900 Cf. A113652, A258277, A258278, A258322.
%Y A257900 Cf. A000122, A000700, A010054, A121373.
%K A257900 sign,mult
%O A257900 1,5
%A A257900 _Michael Somos_, May 25 2015