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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 (list; graph; refs; listen; history; text; internal format)
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", Springer-Verlag, 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] (* Jean-Fran├žois 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

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Last modified January 15 20:47 EST 2019. Contains 319184 sequences. (Running on oeis4.)