A005930 Theta series of D_5 lattice. (Formerly M5270)

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

0,2

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).

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

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.

Example

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

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 *)

Prog

(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 */

Crossrefs

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

Sequence in context: A044178 A044559 A092613 * A261933 A036194 A023695

Adjacent sequences: A005927 A005928 A005929 * A005931 A005932 A005933

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved