A005878 Theta series of cubic lattice with respect to deep hole. (Formerly M4496)

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

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=-oo..oo} q^((2n+1)^2))^3, then divide by q^3 and set q = x^(1/8).

a(n) = 8 * A008443(n).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 2 * exp(3*Pi/8) * Pi^(3/4) * 2^(1/8) / Gamma(3/4)^3 = A388169. - Simon Plouffe, Sep 15 2025

Mathematica

QP = QPochhammer; CoefficientList[(2 QP[q^2]^2/QP[q])^3 + O[q]^63, q] (* Jean-François 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))} \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008
(PARI) {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 A370531

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