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A002706
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Theta series of 6-dimensional lattice A_6^(2) (other names for this lattice or the corresponding quadratic form are LAMBDA_{3,lambda}, P_6^(5), phi_6, F_14).
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1
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1, 0, 42, 56, 84, 168, 280, 336, 462, 336, 840, 672, 1176, 1176, 1386, 1008, 1848, 2016, 2058, 2520, 3528, 2408, 3108, 2688, 4760, 3024, 5880, 4592, 6468, 4704, 5040, 6720, 6930, 6832, 10080, 7224, 7812, 7392, 12600, 7056, 14280, 11760, 12040, 9408
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OFFSET
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0,3
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COMMENTS
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In Elkies 1999 the g.f. is denoted by theta_L. - Michael Somos, Nov 09 2014
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Intro. to 3rd ed.
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). See page 72.
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LINKS
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FORMULA
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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
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EXAMPLE
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G.f. = 1 + 42*q^2 + 56*q^3 + 84*q^4 + 168*q^5 + 280*q^6 + 336*q^7 + 462*q^8 + ...
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MATHEMATICA
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s = (EllipticTheta[3, 0, q] *EllipticTheta[3, 0, q^7] + EllipticTheta[2, 0, q]*EllipticTheta[2, 0, q^7])^3 - 6q*(QPochhammer[q] *QPochhammer[q^7])^3 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 04 2015, from first formula *)
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PROG
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(PARI) {a(n) = local(A, t1, t2, t3); if( n<1, n==0, A = x * O(x^n); t1 = x * (eta(x + A) * eta(x^7 + A))^3; 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 - 6*t1, n))}; /* Michael Somos, Jun 03 2005 */
(Sage) A = ModularForms( Gamma1(7), 3, prec=25) . basis(); (-21*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
(Magma) A := Basis( ModularForms( Gamma1(7), 3), 44); A[1] + 42*A[3] + 56*A[4] + 84*A[5] + 168*A[6] + 280*A[7]; /* Michael Somos, Nov 09 2014 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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