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A226139 Expansion of b(-q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

%I A226139 #16 Feb 16 2025 08:33:19
%S A226139 1,3,-3,-15,-3,18,15,24,-3,-69,-18,36,15,42,-24,-90,-3,54,69,60,-18,
%T A226139 -120,-36,72,15,93,-42,-231,-24,90,90,96,-3,-180,-54,144,69,114,-60,
%U A226139 -210,-18,126,120,132,-36,-414,-72,144,15,171,-93,-270,-42,162,231,216
%N A226139 Expansion of b(-q) * b(q^2) in powers of q where b() is a cubic AGM theta function.
%C A226139 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A226139 Number 21 of the 126 eta-quotients listed in Table 1 of Williams 2012.
%H A226139 G. C. Greubel, <a href="/A226139/b226139.txt">Table of n, a(n) for n = 0..1000</a>
%H A226139 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A226139 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A226139 K. S. Williams, <a href="http://people.math.carleton.ca/~williams/papers/pdf/342.pdf">Fourier series of a class of eta quotients</a>, Int. J. Number Theory 8 (2012), no. 4, 993-1004.
%F A226139 Expansion of = b(q^2) * (2 * b(q^4) - b(q)) = b(q^2)^4 / (b(q) * b(q^4)) in powers of q where b() is a cubic AGM function.
%F A226139 Expansion of (a(q) + a(q^2)) * (a(q^2) - 2 * a(q^4)) / 2 in powers of q where a() is a cubic AGM theta function.
%F A226139 Expansion of (psi(q)^3 / psi(q^3)) * (phi(-q^2)^3 / phi(-q^6)) in power of q where phi(), psi() are Ramanujan theta functions.
%F A226139 Expansion of eta(q^2)^12 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^4) in powers of q.
%F A226139 a(n) = 3 * b(n) where b(n) is multiplicative and b(2^e) = -1, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
%F A226139 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226132.
%F A226139 Euler transform of period 12 sequence [ 3, -9, 2, -6, 3, -6, 3, -6, 2, -9, 3, -4, ...].
%F A226139 G.f.: 1 + 3 * (Sum_{k>0} x^k / (1 - (-x)^k)^2 - 9 * x^(3*k) / (1 - (-x)^(3*k))^2).
%F A226139 G.f.: 1 + 3 * Sum_{k>0 not 2|k} k * (x^k * x^k / (1 + x^(2*k)) - 9 * x^(3*k) / (1 + x^(3*k))).
%F A226139 G.f.: Product_{k>0} (1 - x^(2*k))^12 * (1 + x^(6*k)) / ( (1 - x^k)^3 * (1 + x^(3*k))^3 * (1 - x^(3*k))^2 * (1 - x^(4*k))^3 ).
%F A226139 a(n) = (-1)^n * A131943(n). a(2*n) = A131943(n). a(2*n + 1) = 3 * A134077(n).
%e A226139 1 + 3*q - 3*q^2 - 15*q^3 - 3*q^4 + 18*q^5 + 15*q^6 + 24*q^7 - 3*q^8 + ...
%t A226139 a[ n_] := If[ n < 1, Boole[n == 0], - 3 (-1)^n Sum[ d {0, 1, 0, -2, 0, 1}[[Mod[ d, 6] + 1]], {d, Divisors @n}]]
%t A226139 a[ n_] := If[ n < 2, Boole[n == 0] + 3 Boole[n == 1], 3 Times @@ (Which[ # == 2, -1, # == 3, 4 - 3 #^#2, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])]
%t A226139 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^2]^3 / EllipticTheta[ 4, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)]^3 / EllipticTheta[ 2, 0, q^(3/2)] / 4, {q, 0, n}]
%o A226139 (PARI) {a(n) = if( n<1, n==0, -3 * (-1)^n * sumdiv( n, d, d * [0, 1, 0, -2, 0, 1][d%6 + 1]))}
%o A226139 (PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 3 * prod( k= 1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -1, if( p==3, 4 - p^(e+1), (p^(e+1) - 1) / (p - 1))))))}
%o A226139 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^12 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^4), n))}
%Y A226139 Cf. A131943, A134077, A226132.
%K A226139 sign
%O A226139 0,2
%A A226139 _Michael Somos_, May 27 2013