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A027859 Theta series of odd Leech lattice (the unique unimodular 24-dimensional lattice with minimal norm 3). 2
1, 0, 0, 4096, 98256, 1130496, 8384512, 45785088, 199066704, 726630400, 2314125312, 6606336000, 17213014208, 41497214976, 93722075136, 200284889088, 407539456080, 793560711168, 1487412428800, 2697313996800, 4744432596576, 8110668185600, 13523169792000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

G. Nebe and N. J. A. Sloane, Home page for lattice

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

theta_3(z)^24-48*theta_3(z)^16*Del8(z)+48*theta_3(z)^8*Del8(z)^2 where Del8(z)=[(theta_2(z)*theta_4(z))^4]/16 is the unique cusp form of weight 8 for G(2)... SPLAG p. 187

Expansion of  u * (u^2 - 48*u*v + 48*v^2) in powers of q where u = phi(q)^8, v = q * psi(-q)^8 and phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 01 2017

Expansion of  16*q * f(q)^8 * (u^4 + u^3*v + 3/16*u^2*v^2 + u*v^3 + v^4) / (u * v) in powers of q where u = f(-q)^8, v = 16*q * f(-q^4)^8 and f() is a Ramanujan theta function. - Michael Somos, Feb 01 2017

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 4096 (t/i)^12 f(t) where q = exp(2 Pi i t). - Michael Somos, Feb 01 2017

EXAMPLE

G.f. = 1 + 4096*q^3 + 98256*q^4 + 1130496*q^5 + 8384512*q^6 + ...

MATHEMATICA

a[ n_] := With[ {u = EllipticTheta[ 3, 0, q]^8, v = EllipticTheta[ 2, Pi/4, q^(1/2)]^8}, SeriesCoefficient[ u (u^2 - 3 u v + 3/16 v^2), {q, 0, n}]]; (* Michael Somos, Feb 01 2017 *)

PROG

(PARI) {a(n) = if( n<0, 0, my(A, u, v); A = x * O(x^n); u = eta(x^2 + A)^40 / (eta(x + A) * eta(x^4 + A))^16; v = x * (eta(x + A) * eta(x^4 + A))^8 / eta(x^2 + A)^8; polcoeff( u * (u^2 - 48*u*v + 48*v^2), n))}; /* Michael Somos, Feb 01 2017 */

(PARI) {a(n) = if( n<0, 0, my(A, u, v); A = x * O(x^n); u = eta(x + A)^8; v = 16*x * eta(x^4 + A)^8; polcoeff( 16*x * eta(-x + A)^8 * (u^4 + u^3*v + 3/16*u^2*v^2 + u*v^3 + v^4) / (u * v), n))}; /* Michael Somos, Feb 01 2017 */

(MAGMA) A := Basis( ModularForms( Gamma0(4), 12), 23); A[1] + 4096*A[4] + 98256*A[5] + 1130496*A[6] + 8384512*A[7]; /* Michael Somos, Feb 01 2017 */

CROSSREFS

Sequence in context: A017068 A221489 A017260 * A017368 A016782 A017488

Adjacent sequences:  A027856 A027857 A027858 * A027860 A027861 A027862

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 16 2000

STATUS

approved

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Last modified October 19 13:01 EDT 2019. Contains 328222 sequences. (Running on oeis4.)