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%I #16 Sep 08 2022 08:45:32
%S 1,6,12,14,42,96,84,108,300,278,144,480,546,252,600,672,618,1152,732,
%T 828,2016,1276,720,2112,2100,1302,2040,2078,2100,3360,1872,1740,4908,
%U 3360,1728,4800,5082,2844,4344,4684,3600,6720,4200,3612,10080,5856,3168,8832
%N Expansion of (theta_3(q) * theta_3(q^3))^3 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A135763/b135763.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787871">Mathieu group M24 and modular forms</a>, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
%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))^3 in powers of q where phi() is a Ramanujan theta function.
%F Expansion of (eta(q^2) * eta(q^6))^15 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12))^6 in powers of q.
%F Euler transform of period 12 sequence [ 6, -9, 12, -3, 6, -18, 6, -3, 12, -9, 6, -6, ...].
%F G.f. is a period-1 Fourier series which satisfies f(-1 / (12 t)) = 1728^(1/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
%F G.f.: (( Sum_{k in Z} x^(k^2) ) * ( Sum_{k in Z} x^(3*k^2) ))^3.
%e G.f. = 1 + 6*q + 12*q^2 + 14*q^3 + 42*q^4 + 96*q^5 + 84*q^6 + 108*q^7 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3])^3, {q, 0, n}]; (* _Michael Somos_, Oct 15 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^15 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A))^6, n))};
%o (Magma) A := Basis( ModularForms( Gamma1(12), 3), 48); A[1] + 6*A[2] + 12*A[3] + 14*A[4] + 42*A[5] + 96*A[6] + 84*A[7] + 108*A[8] + 300*A[9] + 278*A[10] + 144*A[11] + 480*A[12] + 546*A[13]; /* _Michael Somos_, Oct 15 2015 */
%K nonn
%O 0,2
%A _Michael Somos_, Nov 28 2007
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