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A005930 Theta series of D_5 lattice.
(Formerly M5270)
3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 6 12:38 EST 2021. Contains 349563 sequences. (Running on oeis4.)