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A008428
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Theta series of D_6 lattice.
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3
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1, 60, 252, 544, 1020, 1560, 2080, 3264, 4092, 4380, 6552, 8160, 8224, 10200, 12480, 14144, 16380, 17400, 18396, 24480, 26520, 23040, 31200, 35904, 32800, 39060, 42840, 44608, 49344, 50520, 54080, 65280, 65532, 57600, 73080, 84864, 74460, 82200, 93600, 92480
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OFFSET
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0,2
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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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. 118.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
G. Nebe and N. J. A. Sloane, Home page for this lattice
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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G.f.: (theta_3(q^(1/2))^6 + theta_4(q^(1/2))^6)/2
Expansion of ( phi(q)^6 + phi(-q)^6 ) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Sep 14 2007
a(n) = A000141(2*n).
G.f. is a period 1 Fourier series that satisfies f(-1 / (8 t)) = 12 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008425. - Michael Somos, Aug 26 2015
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EXAMPLE
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G.f. = 1 + 60*x + 252*x^2 + 544*x^3 + 1020*x^4 + 1560*x^5 + 2080*x^6 + ...
G.f. = 1 + 60*q^2 + 252*q^4 + 544*q^6 + 1020*q^8 + 1560*q^10 + 2080*q^12 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6, {x, 0, 2 n}]; (* Michael Somos, Aug 26 2015 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, 4 * sumdiv(n, d, d^2 * (16 * kronecker(-4, n/d) - kronecker(-4, d))))}; /* Michael Somos, Nov 03 2006 */
(PARI) {a(n) = if( n<0, 0, n*=2; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n))^6, n))}; /* Michael Somos, Nov 03 2006 */
(Magma) A := Basis( ModularForms( Gamma1(8), 3), 80); A[1] + 60*A[3] + 252*A[5] + 544*A[7]; /* Michael Somos, Aug 26 2015 */
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CROSSREFS
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Cf. A000141, A008425.
Sequence in context: A019285 A261970 A206144 * A206232 A075295 A189542
Adjacent sequences: A008425 A008426 A008427 * A008429 A008430 A008431
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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