A125514 Theta series of 4-dimensional lattice QQF.4.i.

1, 4, 20, 4, 52, 24, 20, 32, 116, 4, 120, 48, 52, 56, 160, 24, 244, 72, 20, 80, 312, 32, 240, 96, 116, 124, 280, 4, 416, 120, 120, 128, 500, 48, 360, 192, 52, 152, 400, 56, 696, 168, 160, 176, 624, 24, 480, 192, 244, 228, 620, 72, 728, 216, 20, 288, 928, 80, 600, 240, 312

Offset

0,2

Links

John Cannon, Table of n, a(n) for n = 0..5000

G. Nebe and N. J. A. Sloane, Home page for this lattice

Formula

Contribution from Michael Somos, May 27 2012: (Start)

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 in powers of q.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = + 5*u^4 + 637*v^4 + 1280*w^4 + 352*u^2*w^2 + 342*u^2*v^2 + 5472*v^2*w^2 + 64*u^3*w + 1024*u*w^3 - 68*u^3*v - 756*u*v^3 - 4352*v*w^3 - 3024*v^3*w - 688*u^2*v*w + 2464*u*v^2*w - 2752*u*v*w^2.

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

Convolution of A030188 and A058490. a(3*n) = a(n). (end)

a(n) = 4*b(n) where b(n) is multiplicative and b(2^e) = 2^(e+2) - 3, b(3^e) = 1, b(p^e) = (p^(e+1) - 1)/(p - 1) otherwise. - Michael Somos, Nov 19 2013

a(n) = A006353(n) - A123532(n). a(6*n + 5) = 24 * A098098(n). - Michael Somos, Nov 19 2013

Example

G.f. = 1 + 4*x + 20*x^2 + 4*x^3 + 52*x^4 + 24*x^5 + 20*x^6 + 32*x^7 + 116*x^8 + ...
G.f. = 1 + 4*q^2 + 20*q^4 + 4*q^6 + 52*q^8 + 24*q^10 + 20*q^12 + 32*q^14 + 116*q^16 + ...

Mathematica

a[ n_] := With[{A = QPochhammer[ q] QPochhammer[ q^6], B = QPochhammer[ q^2] QPochhammer[ q^3]}, SeriesCoefficient[ B^7 / A^5 - q A^7 / B^5, {q, 0, n}]] (* Michael Somos, Nov 19 2013 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 2 * qfrep( [ 2, 0, 1, 1; 0, 2, 1, 1; 1, 1, 4, 1; 1, 1, 1, 4 ], n, 1)[n])} /* Michael Somos, May 27 2012 */
(PARI) {a(n) = local(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, n))} /*  Michael Somos, May 27 2012 */
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 4 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 2^(e+2) - 3, if( p==3, 1, (p^(e+1) - 1)/(p - 1))))))} /* Michael Somos, Nov 19 2013 */
(Sage) A = ModularForms( Gamma0(6), 2, prec=56) . basis(); A[0] + 4*A[1] + 20*A[2]; # Michael Somos, Nov 19 2013
(Magma) A := Basis(ModularForms( Gamma0(6), 2)); PowerSeries( A[1] + 4*A[2] + 20*A[3], 56); /* Michael Somos, Nov 19 2013 */

Crossrefs

Cf. A006353, A030188, A058490, A098098, A123532.

Sequence in context: A039921 A081852 A050017 * A118392 A263964 A180855

Adjacent sequences: A125511 A125512 A125513 * A125515 A125516 A125517

Keyword

nonn

Author

N. J. A. Sloane, Jan 31 2007

Status

approved