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A029713 Theta series of 6-dimensional 8-modular lattice of minimal norm 4. 2
1, 0, 30, 56, 66, 144, 188, 288, 378, 448, 528, 504, 884, 1008, 1056, 1440, 1290, 1344, 1834, 1848, 2064, 2880, 2652, 3168, 3332, 2688, 3696, 3696, 4128, 5040, 5280, 5760, 5610, 5824, 6012, 5376, 7798, 8208, 7164, 10080, 8208, 8064, 10560, 8568, 10068 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Associated with permutations in Mathieu group M24 of shape (8)^2(4)(2)(1)^2. - Michael Somos, Nov 24 2007

LINKS

Table of n, a(n) for n=0..44.

M. Koike, Mathieu group M24 and modular forms, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)

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 / (8 t)) = (512)^(1/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 24 2007

A136028(n) = a(n) + 6 * A030207(n). - Michael Somos, Apr 19 2015

EXAMPLE

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

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

MATHEMATICA

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 *)

PROG

(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 */

(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 */

(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 */

CROSSREFS

Cf. A030207, A136028.

Sequence in context: A006315 A027578 A189712 * A154599 A048451 A154796

Adjacent sequences:  A029710 A029711 A029712 * A029714 A029715 A029716

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 16 12:23 EDT 2021. Contains 348041 sequences. (Running on oeis4.)