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A005930
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Theta series of D_5 lattice.
(Formerly M5270)
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1
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1, 40, 90, 240, 200, 560, 400, 800, 730, 1240, 752, 1840, 1200, 2000, 1600, 2720, 1480, 3680, 2250, 3280, 2800, 4320, 2800, 5920, 2960, 5240, 3760, 6720, 4000, 7920, 4800, 6720, 5850, 8960, 4320, 10720, 6200, 9840, 7600, 11040, 5872, 12960, 7520, 12400
(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).
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REFERENCES
| J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| John Cannon, Table of n, a(n) for n = 0..5000
M. Somos, Introduction to Ramanujan theta functions
G. Nebe and N. J. A. Sloane, Home page for this lattice
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| G.f.: (theta_3(q^(1/2))^5+theta_4(q^(1/2))^5)/2
Expansion of ( phi(q)^5 + phi(-q)^5 ) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos Sep 14 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 64 2^(1/2) (t/i)^(5/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A008422.
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EXAMPLE
| 1 + 40*q^2 + 90*q^4 + 240*q^6 + 200*q^8 + 560*q^10 + 400*q^12 + 800*q^14 + ...
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PROG
| (PARI) {a(n)=if(n<0, 0, n*=2; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n))^5, n))} /* Michael Somos Nov 03 2006 */
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CROSSREFS
| A000132(2n)=a(n). A008422 gives dual lattice.
Sequence in context: A044178 A044559 A092613 * A036194 A023695 A038466
Adjacent sequences: A005927 A005928 A005929 * A005931 A005932 A005933
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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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