A008444 Theta series of A_4 lattice.

1, 20, 30, 60, 60, 120, 40, 180, 150, 140, 130, 240, 180, 360, 120, 260, 220, 480, 210, 400, 360, 240, 360, 660, 200, 620, 240, 600, 540, 600, 240, 640, 630, 720, 320, 780, 420, 1080, 600, 480, 650, 840, 360, 1260, 720, 840, 440, 1380, 660, 860, 630, 640, 1080, 1560, 400

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110.

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

Expansion of f(-x)^5 / f(-x^5) + 25 * x * f(-x^5)^5 / f(-x) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Feb 06 2011

Expansion of (1 / Pi) integral_{0 .. Pi/2} theta_3(z, q)^5 + theta_4(z, q)^5 dz in powers of q^2. - Michael Somos, Jan 01 2012

Coefficient of x^0 in the expansion f(x * q, q / x)^5 in powers of q^2 where f() is a Ramanujan theta function. - Michael Somos, Jan 01 2012

G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 5^(3/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A023916. - Michael Somos, Feb 06 2011

A023916(5*n) = a(n) for all n in Z.

Example

G.f. = 1 + 20*x + 30*x^2 + 60*x^3 + 60*x^4 + 120*x^5 + 40*x^6 + 180*x^7 + ...
G.f. = 1 + 20*q^2 + 30*q^4 + 60*q^6 + 60*q^8 + 120*q^10 + 40*q^12 + 180*q^14 + 150*q^16 + 140*q^18 + 130*q^20 + 240*q^22 + 180*q^24 + 360*q^26 + 120*q^28 + 260*q^30 + 220*q^32 + 480*q^34 + 210*q^36 + 400*q^38 + 360*q^40 + 240*q^42 + 360*q^44 + 660*q^46 + 200*q^48 + 620*q^50 + ...

Mathematica

a[ n_] := With[ {u1 = QPochhammer[ x], u5 = QPochhammer[ x^5]}, SeriesCoefficient[ u1^5/u5 + 25 x u5^5/u1, {x, 0, n}]]; (* Michael Somos, Nov 13 2014 *)
terms = 55; f[q_] = LatticeData["A4", "ThetaSeriesFunction"][-I Log[q]/Pi]; s = Series[f[q], {q, 0, 2 terms}]; CoefficientList[s, q^2][[1 ;; terms]] (* Jean-François Alcover, Jul 04 2017 *)

Prog

(Magma) L := Lattice("A", 4); A<q> := ThetaSeries(L, 120); A;
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 / eta(x^5 + A) + 25 * x * eta(x^5 + A)^5 / eta(x + A), n))}; /* Michael Somos, Feb 06 2011 */
(Magma) A := Basis( ModularForms( Gamma1(5), 2), 55) ; A[1] + 20*A[2] + 30*A[3]; /* Michael Somos, Nov 13 2014 */

Crossrefs

Cf. A000007, A000122, A004016, A004015, A008445, A008446, A008447, A008448, A008449 (Theta series of lattices A_0, A_1, A_2, A_3, A_5, ...).

Sequence in context: A066027 A256227 A142342 * A268984 A359002 A066214

Adjacent sequences: A008441 A008442 A008443 * A008445 A008446 A008447

Keyword

nonn,nice

Author

N. J. A. Sloane

Status

approved