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%I #15 Mar 12 2021 22:24:47
%S 1,-4,-12,-4,-12,-24,-12,-32,-12,-4,-72,-48,-12,-56,-96,-24,-12,-72,
%T -12,-80,-72,-32,-144,-96,-12,-124,-168,-4,-96,-120,-72,-128,-12,-48,
%U -216,-192,-12,-152,-240,-56,-72,-168,-96,-176,-144,-24,-288,-192,-12,-228
%N Expansion of (3 * phi(q^3)^4 - phi(q)^4) / 2 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).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A230057/b230057.txt">Table of n, a(n) for n = 0..1000</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^3) * a(-q) in powers of q where a() is a cubic AGM theta function.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = -12 * (t/i)^2 f(t) where q = exp(2 Pi i t).
%F a(n) = -4 * b(n) where b(n) is multiplicative and b(2^e) = 3 if e>0, b(3^e) = 1, b(p^e) = (p^(e+1) - 1)/(p - 1), if p>3. - _Michael Somos_, Nov 19 2013
%F G.f.: 1 - 4 * ( 1*x^1 / (1 + (-x)^1) + 2*x^2 / (1 + (-x)^2) + 4*x^4 / (1 + (-x)^4) + 5*x^5 / (1 + (-x)^5) + 7*x^7 / (1 + (-x)^7) + ...).
%F a(2*n) = a(4*n) = A186100(n). a(3*n) = a(n).
%e G.f. = 1 - 4*q - 12*q^2 - 4*q^3 - 12*q^4 - 24*q^5 - 12*q^6 - 32*q^7 + ...
%t a[ n_]:= SeriesCoefficient[(3*EllipticTheta[3,0,q^3]^4 - EllipticTheta[ 3,0,q]^4)/2, {q, 0, n}];
%o (PARI) {a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d * (2 + (-1)^(n/d)) * kronecker( 36, d)))}
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (3 * sum( k=1, sqrtint(n\3), 2 * x^(3*k^2), 1 + A)^4 - sum( k=1, sqrtint(n), 2 * x^k^2, 1 + A)^4) / 2, n))}
%o (PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); -4 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 3, if( p==3, 1, (p^(e+1) - 1)/(p - 1))))))} /* _Michael Somos_, Nov 19 2013 */
%Y Cf. A186100.
%K sign
%O 0,2
%A _Michael Somos_, Oct 07 2013
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