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%I #30 Sep 08 2022 08:44:31
%S 1,6,24,56,114,168,280,294,444,390,840,636,1176,1176,1512,1008,1782,
%T 2016,1896,2520,3528,2408,3216,2796,4760,3174,5880,4592,6258,4380,
%U 5040,6720,7200,6832,10080,7224,8082,7164,12600,7056,14280,11760,12040,9756
%N Expansion of (theta_3(z)*theta_3(7z)+theta_2(z)*theta_2(7z))^3.
%C Theta series of Kleinian lattice (Z[ (-1+sqrt(-7))/2 ])^3 in 3 complex (or 6 real) dimensions.
%D N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103)
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53.
%H Robert Israel, <a href="/A002653/b002653.txt">Table of n, a(n) for n = 0..10000</a>
%H H. H. Chan and Y. L. Ong, <a href="https://doi.org/10.1090/S0002-9939-99-04832-7">On Eisenstein series and Sum_{m,n} q^(m^2+mn+2n^2)</a>, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1735-1744, See page 1737. MR1600120 (99i:11029).
%H N. Elkies, <a href="http://www.msri.org/publications/books/Book35/files/elkies.pdf">The Klein quartic in number theory</a>
%F G.f.: (theta_3(z)*theta_3(7*z) + theta_2(z)*theta_2(7*z))^3.
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) is a homogeneous degree 6 polynomial with 28 terms. - _Michael Somos_, Jun 03 2005
%F Expansion of (eta(q)^8 + 13 * eta(q)^4 * eta(q^7)^4 + 49 * eta(q^7)^8) / ( eta(q) * eta(q^7) ) in power of q. - _Michael Somos_, Mar 11 2008
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (7*t)) = 7^(3/2)*(t/i)^3*f(t) where q = exp(2*Pi*i*t).
%e G.f. = 1 + 6*q + 24*q^2 + 56*q^3 + 114*q^4 + 168*q^5 + 280*q^6 + 294*q^7 + ...
%p g:= (JacobiTheta3(0,z) * JacobiTheta3(0,z^7) + JacobiTheta2(0,z) * JacobiTheta2(0,z^7))^3:
%p S:= series(g,z,101):
%p seq(coeff(S,z,i),i=0..100); # _Robert Israel_, Aug 12 2020
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^7] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^7])^3, {q, 0, n}]; (* _Michael Somos_, Nov 09 2014 *)
%o (PARI) {a(n) = local(A, t2, t3); if( n<1, n==0, A = x * O(x^n); t2 = sum(k=1, (sqrtint(4*n + 1) + 1)\2, 2*x^(k*k - k), A); t3 = sum(k=1, sqrtint(n), 2*x^(k*k), 1 + A); A = x * O(x^(n\7)); polcoeff( (t3 * subst(t3 + A, x, x^7) + x^2 * t2 * subst(t2 + A, x, x^7))^3, n))}; /* _Michael Somos_, Jun 03 2005 */
%o (PARI) {a(n) = local(A, t1, t7); if( n<0, 0, A = x * O(x^n); t1 = eta(x + A)^4; t7 = eta(x^7 + A)^4; polcoeff( (t1^2 + 13 * x * t1 * t7 + 49 * x^2 * t7^2) / (t1 * t7)^(1/4), n))}; /* _Michael Somos_, Mar 11 2008 */
%o (Sage) A = ModularForms( Gamma1(7), 3, prec=60) . basis(); (3*A[0] + 4*A[1] + 21*A[2] + 105*A[3] + 224*A[4] + 441*A[5] + 672*A[6])/4 # _Michael Somos, May 25 2014
%o (Magma) A := Basis( ModularForms( Gamma1(7), 3), 44); A[1] + 6*A[2] + 24*A[3] + 56*A[4] +114*A[5] + 168*A[6] + 280*A[7]; /* _Michael Somos_, Nov 09 2014 */
%Y Cf. A002652.
%K nonn,look
%O 0,2
%A _N. J. A. Sloane_
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