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A008654
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Theta series of direct sum of 3 copies of hexagonal lattice.
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0
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1, 18, 108, 234, 234, 864, 756, 900, 1836, 2178, 1296, 4320, 3042, 3060, 5400, 6048, 3690, 10368, 6588, 6516, 11232, 11700, 6480, 19008, 12852, 10818, 18360, 19674, 11700, 30240, 16848, 17316, 29484, 30240, 15552, 43200, 28314, 24660, 39096
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
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REFERENCES
| J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110.
B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 124, Equation (7.19).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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FORMULA
| Expansion of (theta_3(z)*theta_3(3z)+theta_2(z)*theta_2(3z))^3.
Expansion of a(q)^3 in powers of q where a() is a cubic AGM function. - Michael Somos, Sep 04 2008
Expansion of (eta(q)^12 + 27 * eta(q^3)^12) / (eta(q) * eta(q^3))^3 in powers of q. - Michael Somos, Sep 04 2008
Expansion of (f(-q)^12 + 27 * q * f(-q^3)^12) / (f(-q) * f(-q^3))^3 in powers of q where f() is a Ramanujan theta function. - Michael Somos, Sep 04 2008
G.f. is a period 1 Fourier series which satisfies f(-1/(3 t)) = 3^(3/2) (t / i)^3 f(t) where q = exp(2 pi i t). - Michael Somos, Sep 04 2008
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EXAMPLE
| 1 + 18*q + 108*q^2 + 234*q^3 + 234*q^4 + 864*q^5 + 756*q^6 + 900*q^7 + ...
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PROG
| (PARI) {a(n) = local(A, A3); if( n<0, 0, A = x * O(x^n); A3 = eta(x^3 + A)^3; A = eta(x + A)^3; polcoeff( (A^4 + 27 * x * A3^4) / (A * A3), n))} /* Michael Somos Sep 04 2008 */ - Michael Somos, Sep 04 2008
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CROSSREFS
| Sequence in context: A123277 A123595 A002165 * A060787 A019584 A041622
Adjacent sequences: A008651 A008652 A008653 * A008655 A008656 A008657
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Michael Somos, Sep 04 2008
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