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A125509 Theta series of 4-dimensional lattice QQF.4.f. 2
1, 0, 0, 12, 12, 0, 12, 0, 24, 12, 24, 24, 24, 12, 24, 24, 24, 24, 36, 24, 48, 48, 48, 0, 60, 48, 36, 36, 72, 36, 96, 12, 60, 24, 48, 48, 108, 48, 72, 84, 96, 60, 120, 48, 72, 72, 0, 60, 132, 48, 96, 96, 120, 48, 132, 96, 144, 72, 108, 48, 168, 96, 108, 96, 144, 72, 144, 72, 144, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The quadratic form associated with the lattice can be expressed as Q(x, y, z, w) = (2*x+z+w)^2 + (x+2*y)^2 + (x+y)^2 + (y+2*z)^2 + (z+2*w)^2 + w^2 which comes from the basis (2,1,1,0,0,0), (0,2,1,1,0,0), (1,0,0,2,1,0), (1,0,0,0,2,1). - Michael Somos, Mar 30 2015

LINKS

John Cannon, Table of n, a(n) for n = 0..5000

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

EXAMPLE

G.f. = 1 + 12*x^3 + 12*x^4 + 12*x^6 + 24*x^8 + 12*x^9 + 24*x^10 + 24*x^11 + ...

G.f. = 1 + 12*q^6 + 12*q^8 + 12*q^12 + 24*q^16 + 12*q^18 + 24*q^20 + 24*q^22 + 24*q^24 + ...

MATHEMATICA

a[ n_] := With[{B = QPochhammer[ x^2] QPochhammer[ x^46]}, With[{A = QPochhammer[ x] QPochhammer[ x^23] / B}, SeriesCoefficient[ (4 x^3 + 4 A x^2 + A^3) (4 x^3 + 4 A x^2 + 4 A^2 x + A^3) (B / A)^2, {x, 0, n}]]]; (* Michael Somos, Mar 23 2015 *)

PROG

(PARI)  {a(n) = my(G); if( n<0, 0, G = [ 6, 3, 2, 2; 3, 6, 2, 0; 2, 2, 6, 3; 2, 0, 3, 6 ]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n, 1)), n))}; /* Michael Somos, Mar 23 2015 */

(PARI) {a(n) = my(A, B); if( n<0, 0, A = x * O(x^n); B = eta(x^2 + A) * eta(x^46 + A); A = eta(x + A) * eta(x^23 + A) / B; polcoeff( (4*x^3 + 4*A*x^2 + A^3) * (4*x^3 + 4*A*x^2 + 4*A^2*x + A^3) * (B / A)^2, n))}; /* Michael Somos, Mar 23 2015 */

(MAGMA) Basis( ModularForms( Gamma0(23), 2), 70)[1]; /* Michael Somos, Mar 23 2015 */

(Sage) A = ModularForms( Gamma0(23), 2, prec=70).basis(); A[2] - 12/11*(A[0] + 3*A[1]); # Michael Somos, Mar 23 2015

CROSSREFS

Sequence in context: A195748 A038337 A155825 * A281251 A247511 A097824

Adjacent sequences:  A125506 A125507 A125508 * A125510 A125511 A125512

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 31 2007

STATUS

approved

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Last modified January 22 22:16 EST 2020. Contains 331166 sequences. (Running on oeis4.)