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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

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

Examples

			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 + ...
		

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

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, ...).

Programs

Magma

L := Lattice("A", 4); A := ThetaSeries(L, 120); A;

Magma

A := Basis( ModularForms( Gamma1(5), 2), 55) ; A[1] + 20*A[2] + 30*A[3]; /* Michael Somos, Nov 13 2014 */

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 *)

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 */

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.