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A004008
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Expansion of theta series of E_7 lattice in powers of q^2.
(Formerly M5388)
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3
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1, 126, 756, 2072, 4158, 7560, 11592, 16704, 24948, 31878, 39816, 55944, 66584, 76104, 99792, 116928, 133182, 160272, 177660, 205128, 249480, 265104, 281736, 350784, 382536, 390726, 470232, 505568, 532800, 615384, 640080, 701568, 799092, 809424, 853776
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OFFSET
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0,2
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COMMENTS
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125. Equation (112)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Expansion of phi(q)^3 * (phi(q)^4 + 7 * 16 * q * psi(q^2)^4) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 24 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(1/2) (t / i)^(7/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A003781. - Michael Somos, Aug 27 2013
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EXAMPLE
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G.f. = 1 + 126*x + 756*x^2 + 2072*x^3 + 4158*x^4 + 7560*x^5 + 11592*x^6 + ...
G.f. = 1 + 126*q^2 + 756*q^4 + 2072*q^6 + 4158*q^8 + 7560*q^10 + 11592*q^12 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 ( 8 EllipticTheta[ 3, 0, q]^4 - 7 EllipticTheta[ 4, 0, q]^4), {q, 0, n}]; (* Michael Somos, Aug 27 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 ( EllipticTheta[ 3, 0, q]^4 + 7 EllipticTheta[ 2, 0, q]^4), {q, 0, n}]; (* Michael Somos, Apr 21 2015 *)
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PROG
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(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^3 * (8 * A^4 - 7 * subst(A, x, -x)^4), n))}; /* Michael Somos, Oct 24 2006 */
(PARI) {a(n) = my(G); if( n<1, n==0, G = [2, -1, 0, 0, 0, 0, 0; -1, 2, -1, 0, 0, 0, 0; 0, -1, 2, -1, 0, 0, 0; 0, 0, -1, 2, -1, 0, -1; 0, 0, 0, -1, 2, -1, 0; 0, 0, 0, 0, -1, 2, 0; 0, 0, 0, -1, 0, 0, 2]; 2 * qfrep( G, n, 1)[n])}; /* Michael Somos, Jun 11 2007 */
(Magma) A := Basis( ModularForms( Gamma0(4), 7/2), 50); A[1] + 126*A[2]; /* Michael Somos, Jun 09 2014 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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