

A005878


Theta series of cubic lattice with respect to deep hole.
(Formerly M4496)


6



8, 24, 24, 32, 48, 24, 48, 72, 24, 56, 72, 48, 72, 72, 48, 48, 120, 72, 56, 96, 24, 120, 120, 48, 96, 96, 72, 96, 120, 48, 104, 168, 96, 48, 120, 72, 96, 192, 72, 144, 96, 72, 144, 120, 96, 104, 192, 72, 120, 192, 48, 144, 216, 48, 96, 120, 144, 192, 168, 120, 96, 216, 72
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OFFSET

0,1


COMMENTS

Number of ways of writing 8*n+3 as the sum of three odd squares.  Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008
Expansion of Jacobi theta constant theta_2^3.  Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008


REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", SpringerVerlag, p. 107.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..99 from Herman Jamke (hermanjamke(AT)fastmail.fm))
G. Nebe and N. J. A. Sloane, Home page for this lattice


FORMULA

G.f.: Form (Sum_{n=inf..inf} q^((2n+1)^2))^3, then divide by q^3 and set q = x^(1/8).
a(n) = 8 * A008443(n).


MATHEMATICA

QP = QPochhammer; CoefficientList[(2 QP[q^2]^2/QP[q])^3 + O[q]^63, q] (* JeanFrançois Alcover, Jul 04 2017 *)


PROG

(PARI) {a(n)=if(n<0, 0, 8*polcoeff( sum(k=0, (sqrtint(8*n+1)1)\2, x^((k^2+k)/2), x*O(x^n))^3, n))} {a(n)=local(A); if(n<0, 0, A=x*O(x^n); 8*polcoeff( (eta(x^2+A)^2/eta(x+A))^3, n))} \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008


CROSSREFS

Equals 8 times A008443. Cf. A085121.
Sequence in context: A263630 A205376 A088448 * A128637 A280708 A109272
Adjacent sequences: A005875 A005876 A005877 * A005879 A005880 A005881


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008


STATUS

approved



