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A005879 Theta series of D_4 lattice with respect to deep hole.
(Formerly M4509)
4
8, 32, 48, 64, 104, 96, 112, 192, 144, 160, 256, 192, 248, 320, 240, 256, 384, 384, 304, 448, 336, 352, 624, 384, 456, 576, 432, 576, 640, 480, 496, 832, 672, 544, 768, 576, 592, 992, 768, 640, 968, 672, 864, 960, 720, 896, 1024, 960, 784, 1248, 816, 832, 1536 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The D_4 lattice is the set of all integer quadruples [a, b, c, d] where a + b + c + d is even. The deep holes are quadruples [a, b, c, d] where each coordinate is half an odd integer and where a + b + c + d is even. - Michael Somos, May 23 2102

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).

LINKS

T. D. Noe, Table of n, a(n) for n=0..10000

G. Nebe and N. J. A. Sloane, Home page for this lattice

Index entries for sequences related to D_4 lattice

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of Jacobi theta_2(q)^4/(2q) in powers of q^2. - Michael Somos, Apr 11 2004

Expansion of q^(-1/2) * 8 * (eta(q^2)^2 / eta(q))^4 in powers of q. - Michael Somos, Apr 11 2004

Expansion of 8 * psi(x)^4 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, May 23 2012

Expansion of (phi(q)^4 - phi(-q)^4) / (2 * q) in powers of q^2. - Michael Somos, May 23 2012

G.f.: 8 * (Product_{k>0} (1 - x^k) * (1 + x^k)^2)^4. - Michael Somos, Apr 11 2004

a(n) = 8 * A008438(n) = 4 * A005880(n) = A000118(2*n + 1) = - A096727(2*n + 1). - Michael Somos, Nov 01 2006

EXAMPLE

8 + 32*x + 48*x^2 + 64*x^3 + 104*x^4 + 96*x^5 + 112*x^6 + 192*x^7 + ...

8*q + 32*q^3 + 48*q^5 + 64*q^7 + 104*q^9 + 96*q^11 + 112*q^13 + ...

PROG

(PARI) {a(n) = if( n<0, 0, 8 * sigma(2*n + 1))} /* Michael Somos, Apr 11 2004 */

CROSSREFS

Cf. A000118, A005880, A008438, A096727.

Sequence in context: A144096 A127988 A129749 * A067519 A009245 A018842

Adjacent sequences:  A005876 A005877 A005878 * A005880 A005881 A005882

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 23 22:11 EDT 2014. Contains 240947 sequences.