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A028996
Theta series of quadratic form with Gram matrix [ 4, 1, 0, 2; 1, 4, 2, 0; 0, 2, 4, -1; 2, 0, -1, 4 ].
2
1, 0, 12, 12, 12, 12, 24, 24, 36, 36, 48, 0, 72, 24, 48, 60, 84, 48, 84, 48, 96, 72, 12, 60, 144, 84, 120, 84, 144, 72, 168, 60, 132, 12, 120, 120, 228, 84, 144, 144, 216, 120, 240, 120, 12, 192, 168, 96, 288, 144, 204, 168, 216, 144, 312, 12, 288, 192, 216, 132, 408, 120, 264
OFFSET
0,3
LINKS
G. Nebe and N. J. A. Sloane, Home page for this lattice
FORMULA
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, May 10 2015
EXAMPLE
G.f. = 1 + 12*x^2 + 12*x^3 + 12*x^4 + 12*x^5 + 24*x^6 + 24*x^7 + 36*x^8 + ...
G.f. = 1 + 12*q^4 + 12*q^6 + 12*q^8 + 12*q^10 + 24*q^12 + 24*q^14 + 36*q^16 + ...
MATHEMATICA
a[ n_] := With[ {A1 = q (QPochhammer[ q] QPochhammer[ q^11])^2, A2 = q^2 (QPochhammer[ q^2] QPochhammer[ q^22])^2}, SeriesCoefficient[ A1^2 / A2 + 4 * (A1 + A2 + A2^2 / A1), {q, 0, 2n}]]; (* Michael Somos, Aug 27 2014 *)
PROG
(Magma) T := ThetaSeries(Lattice(LatticeDatabase(), "QQF.4.b"), 120) ; [ Coefficient(T, n) : n in [ k : k in [0..120] | IsEven(k) ] ] ; /* Klaus Brockhaus, Feb 01 2007 */
(Magma) [ Coefficient(Basis(ModularForms(Gamma0(11), 2))[1], n) : n in [1..100] ]; /* Klaus Brockhaus, Feb 01 2007 */
(PARI) {a(n) = my(A, A1, A2); if( n<0, 0, n *= 2; A = x * O(x^n); A1 = x * (eta(x + A) * eta(x^11 + A))^2; A2 = x^2 * (eta(x^2 + A) * eta(x^22 + A))^2; polcoeff( A1^2 / A2 + 4 * (A1 + A2 + A2^2 / A1), n))}; /* Michael Somos, Aug 27 2014 */
(PARI) {a(n) = my(G); if( n<0, 0, G = [ 4, 1, 0, 2; 1, 4, 2, 0; 0, 2, 4, -1; 2, 0, -1, 4]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n, 1)), n))}; /* Michael Somos, May 10 2015 */
CROSSREFS
Sequence in context: A122878 A302341 A064162 * A113595 A109053 A186100
KEYWORD
nonn
AUTHOR
STATUS
approved