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theta3(z) * theta3(2*z) * theta3(3*z) * theta3(6*z).
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%I #24 Sep 08 2022 08:44:51

%S 1,2,2,6,6,4,14,8,6,26,12,16,42,12,16,44,6,20,50,16,36,56,24,16,42,30,

%T 28,78,48,36,84,40,6,80,36,48,150,44,40,100,36,36,112,48,72,148,48,48,

%U 42,50,62,124,84,52,158,64,48,144,60,64,252,60,64,200,6,88,168,64,108

%N theta3(z) * theta3(2*z) * theta3(3*z) * theta3(6*z).

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102, eq. 9.

%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 225.

%H Seiichi Manyama, <a href="/A033712/b033712.txt">Table of n, a(n) for n = 0..10000</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 Number of solutions to a^2 + 2*b^2 + 3*c^2 + 6*d^2 = n in integers.

%F Expansion of phi(q) * phi(q^2) * phi(q^3) * phi(q^6) in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Apr 19 2015

%F Expansion of (eta(q^2) * eta(q^4) * eta(q^6) * eta(q^12))^3 / (eta(q) * eta(q^3) * eta(q^8) * eta(q^24))^2 in powers of q.

%F Euler transform of period 24 sequence [2, -1, 4, -4, 2, -2, 2, -2, 4, -1, 2, -8, 2, -1, 4, -2, 2, -2, 2, -4, 4, -1, 2, -4, ...]. - _Michael Somos_, May 30 2005

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24 (t/i)^2 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Apr 19 2015

%F a(2*n) = A282544(n). a(4*n) = A125510(n).

%e G.f. = 1 + 2*q + 2*q^2 + 6*q^3 + 6*q^4 + 4*q^5 + 14*q^6 + 8*q^7 + 6*q^8 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3] EllipticTheta[ 3, 0, q^6], {q, 0, n}]; (* _Michael Somos_, Apr 19 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A * subst( A + x * O(x^(n\2)), x ,x^2) * subst( A + x * O(x^(n\3)), x, x^3) * subst( A + x * O(x^(n\6)), x, x^6), n))}; /* _Michael Somos_, May 30 2005 */

%o (PARI) {a(n) = my(G); if( n<0, 0, G = [1, 0, 0, 0; 0, 2, 0, 0; 0, 0, 3, 0; 0, 0, 0, 6 ]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* _Michael Somos_, Apr 19 2015 */

%o (Magma) A := Basis( ModularForms( Gamma0(24), 2), 69); A[1] + 2*A[2] + 2*A[3] + 6*A[4] + 6*A[5] + 4*A[6] + 14*A[7] + 6*A[8]; /* _Michael Somos_, Apr 19 2015 */

%Y Cf. A125510, A282544.

%K nonn

%O 0,2

%A _N. J. A. Sloane_