A005930 Theta series of D_5 lattice.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Examples

			1 + 40*q^2 + 90*q^4 + 240*q^6 + 200*q^8 + 560*q^10 + 400*q^12 + 800*q^14 + ...
		

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

• John Cannon, Table of n, a(n) for n = 0..5000
• G. Nebe and N. J. A. Sloane, Home page for this lattice
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

A000132(2n) = a(n). A008422 gives dual lattice.

Programs

Mathematica

terms = 44; phi[q_] := EllipticTheta[3, 0, q]; s = (phi[q]^5 + phi[-q]^5)/2 + O[q]^(2 terms); DeleteCases[CoefficientList[s, q], 0][[1 ;; terms]] (* Jean-François Alcover, Jul 04 2017, after Michael Somos *)

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 */

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.