A029713 - OEIS

%I #13 Sep 08 2022 08:44:50

%S 1,0,30,56,66,144,188,288,378,448,528,504,884,1008,1056,1440,1290,

%T 1344,1834,1848,2064,2880,2652,3168,3332,2688,3696,3696,4128,5040,

%U 5280,5760,5610,5824,6012,5376,7798,8208,7164,10080,8208,8064,10560,8568,10068

%N Theta series of 6-dimensional 8-modular lattice of minimal norm 4.

%C Associated with permutations in Mathieu group M24 of shape (8)^2(4)(2)(1)^2. - _Michael Somos_, Nov 24 2007

%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787871">Mathieu group M24 and modular forms</a>, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/6QF.6.d.html">Home page for this lattice</a>

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = (512)^(1/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Nov 24 2007

%F A136028(n) = a(n) + 6 * A030207(n). - _Michael Somos_, Apr 19 2015

%e G.f. = 1 + 30*x^2 + 56*x^3 + 66*x^4 + 144*x^5 + 188*x^6 + 288*x^7 + ...

%e G.f. = 1 + 30*q^4 + 56*q^6 + 66*q^8 + 144*q^10 + 188*q^12 + 288*q^14 + ...

%t a[ n_] := SeriesCoefficient[ With[{e1 = QPochhammer[ x] QPochhammer[ x^8], e2 = QPochhammer[ x^2] QPochhammer[ x^4]}, e2^9 / e1^6 - 6 x e1^2 e2], {x, 0, n}]; (* _Michael Somos_, Apr 19 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A) * eta(x^4 + A) )^9 / ( eta(x + A) * eta(x^8 + A) )^6 - 6 * x * ( eta(x + A) * eta(x^8 + A) )^2 * eta(x^2 + A) * eta(x^4 + A), n))}; /* _Michael Somos_, Nov 24 2007 */

%o (PARI) {a(n) = my(G); if( n<0, 0, G = [4, 1, -1, -1, 1, -1; 1, 4, 0, 1, 2, 1; -1, 0, 4, -1, 2, -1; -1, 1, -1, 4, -1, 0; 1, 2, 2, -1, 4, -1; -1, 1, -1, 0, -1, 4]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n, 1)), n))}; /* _Michael Somos_, Nov 24 2007 */

%o (Magma) A := Basis( ModularForms( Gamma1(8), 3), 45); A[1] + 30*A[3] + 56*A[4] + 66*A[5] + 144*A[6] + 188*A[7]; /* _Michael Somos_, Apr 19 2015 */

%Y Cf. A030207, A136028.

%K nonn

%O 0,3

%A _N. J. A. Sloane_.