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%I #15 Mar 12 2021 22:24:46
%S 1,-1,-3,5,-3,-6,15,-8,-3,23,-18,-12,15,-14,-24,30,-3,-18,69,-20,-18,
%T 40,-36,-24,15,-31,-42,77,-24,-30,90,-32,-3,60,-54,-48,69,-38,-60,70,
%U -18,-42,120,-44,-36,138,-72,-48,15,-57,-93,90,-42,-54,231,-72,-24,100
%N Expansion of b(q^2) * (b(q) + 2 * b(q^4)) / 3 in powers of q where b() is a cubic AGM 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="/A214456/b214456.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 (9 * phi(q^3)^4 - phi(q)^4) / 8 = phi(q) * (psi(-q) * chi(q^3))^3 = psi(-q)^4 * (chi(q) * chi(q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
%F Expansion of eta(q) * eta(q^2)^2 * eta(q^4) * eta(q^6)^6 / ( eta(q^3) * eta(q^12))^3 in powers of q.
%F Euler transform of period 12 sequence [ -1, -3, 2, -4, -1, -6, -1, -4, 2, -3, -1, -4, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 36 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A214361.
%F a(n) = -b(n) where b() is multiplicative with b(2^e) = 3 if e>0, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
%F G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 + x^(6*k-3))^3 / (1 + x^(2*k-1)).
%F a(n) = (-1)^n * A118271(n). a(2*n) = a(4*n) = A131943(n). a(2*n + 1) = -A134077(n).
%e 1 - q - 3*q^2 + 5*q^3 - 3*q^4 - 6*q^5 + 15*q^6 - 8*q^7 - 3*q^8 + 23*q^9 + ...
%t a[ n_] := SeriesCoefficient[ (9 EllipticTheta[ 3, 0, q^3]^4 - EllipticTheta[ 3, 0, q]^4) / 8, {q, 0, n}]
%t a[ n_] := SeriesCoefficient[ QPochhammer[q^2]^4 QPochhammer[-q^3, q^6]^3 /QPochhammer[-q, q^2], {q, 0, n}]
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] QPochhammer[ -q^3, q^6])^3 EllipticTheta[ 2, 0, (-q)^(1/2)]^4 / (16 (-q)^(1/2)), {q, 0, n}]
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^2 * eta(x^4 + A) * eta(x^6 + A)^6 / ( eta(x^3 + A) * eta(x^12 + A))^3, n))}
%o (PARI) {a(n) = if( n<1, n==0, -sigma(n) + if( n%3==0, 9 * sigma(n/3)) + if( n%4==0, 4 * sigma(n/4)) + if( n%12==0, -36 * sigma(n/12)))}
%o (PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 3, if( p==3, 4 - 3^(e+1), (p^(e+1) - 1) / (p - 1))))))}
%Y Cf. A118271, A131943, A134077, A214361.
%K sign
%O 0,3
%A _Michael Somos_, Jul 18 2012
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