A212817 Theta series of direct sum of 2 copies of 4-dimensional lattice QQF.4.i.

1, 8, 56, 168, 536, 624, 2328, 1600, 4184, 4872, 7824, 6432, 19320, 10672, 21568, 22320, 33752, 23184, 62904, 32992, 66000, 61248, 83040, 58944, 155832, 75320, 136912, 130728, 179776, 117168, 291024, 142720, 269528, 236448, 307440, 207744, 528024, 243952

Offset: 0

Author

Michael Somos, May 27 2012

Keywords

• nonn

Examples

			G.f. = 1 + 8*x + 56*x^2 + 168*x^3 + 536*x^4 + 624*x^5 + 2328*x^6 + 1600*x^7 + ...
		

Links

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

Crossrefs

Cf. A028977, A030209, A125514.

Programs

Magma

A := Basis( ModularForms( Gamma0(6), 4), 38); A[1] + 8*A[2] + 56*A[3] + 168*A[4] + 536*A[5]; /* Michael Somos, Jun 04 2015 */

Mathematica

a[ n_] := SeriesCoefficient[ With[{e1 = QPochhammer[ x] QPochhammer[ x^6], e2 = QPochhammer[ x^2] QPochhammer[ x^3]}, (e2^7 / e1^5 - x e1^7 / e2^5)^2 ], {x, 0, n}]; (* Michael Somos, Apr 19 2015 *)

PARI

{a(n) = my(A, B); if( n<0, 0, A = x * O(x^n); B = eta(x^2 + A) * eta(x^3 + A); A = eta(x + A) * eta(x^6 + A); polcoeff( (B^7 / A^5 - x * A^7 / B^5)^2, n))};

PARI

{a(n) = my(G); if( n<0, 0, G = [ 2, 0, 1, 1; 0, 2, 1, 1; 1, 1, 4, 1; 1, 1, 1, 4 ]; polcoeff( (1 + 2 * x * Ser( qfrep( G, n, 1)))^2, n))};

Formula

Expansion of ((eta(q^2) * eta(q^3))^7 / (eta(q) * eta(q^6))^5 - (eta(q) * eta(q^6))^7 / (eta(q^2) * eta(q^3))^5)^2 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 576 (t/i)^4 f(t) where q = exp(2 Pi i t).

Convolution square of A125514.

a(n) = A028977(n) + 8 * A030209(n). - Michael Somos, Jun 05 2015