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

0,2

Links

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

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

Example

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

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 *)

Prog

(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))};
(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 */

Crossrefs

Cf. A028977, A030209, A125514.

Sequence in context: A364124 A044146 A118772 * A355948 A267522 A073831

Adjacent sequences: A212814 A212815 A212816 * A212818 A212819 A212820

Keyword

nonn

Author

Michael Somos, May 27 2012

Status

approved