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
Keywords
Examples
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 + ...
Links
- John Cannon, Table of n, a(n) for n = 0..5000
- G. Nebe and N. J. A. Sloane, Home page for this lattice
Programs
-
Magma
A := Basis(ModularForms( Gamma0(6), 2)); PowerSeries( A[1] + 4*A[2] + 20*A[3], 56); /* Michael Somos, Nov 19 2013 */
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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 *) -
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
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).
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
Comments