A028887 Theta series of 4-dimensional 5-modular lattice with det 25 and minimal norm 2.

1, 6, 18, 24, 42, 6, 72, 48, 90, 78, 18, 72, 168, 84, 144, 24, 186, 108, 234, 120, 42, 192, 216, 144, 360, 6, 252, 240, 336, 180, 72, 192, 378, 288, 324, 48, 546, 228, 360, 336, 90, 252, 576, 264, 504, 78, 432, 288, 744, 342, 18, 432, 588, 324, 720, 72, 720, 480

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			G.f. = 1 + 6*x + 18*x^2 + 24*x^3 + 42*x^4 + 6*x^5 + 72*x^6 + 48*x^7 + ...
G.f. = 1 + 6*q^2 + 18*q^4 + 24*q^6 + 42*q^8 + 6*q^10 + 72*q^12 + 48*q^14 + ...
		

References

• B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 463 Entry 4(i).

Links

• John Cannon, Table of n, a(n) for n = 0..5000
• Shaun Cooper and Dongxi Ye, Level 14 AND 15 Analogues of Ramanujan's Elliptic Functions to Alternative Bases, preprint, 2015.
• G. Nebe and N. J. A. Sloane, Home page for this lattice

Programs

Magma

Basis( ModularForms( Gamma0(5), 2), 70) [1]; /* Michael Somos, Jun 12 2014 */

Mathematica

a[ n_] := If[ n < 1, Boole[ n == 0], 6 Sum[ If[ Mod[ d, 5] > 0, d, 0], {d, Divisors @ n }]];  (* Michael Somos, Jun 12 2014 *)
a[ n_] := SeriesCoefficient[ 1 + 6 Sum[ k x^k / (1 - x^k) - 5 k x^(5 k) / (1 - x^(5 k)), {k, n}], {x, 0, n}]; (* Michael Somos, Jun 12 2014 *)

PARI

{a(n) = if( n<1, n==0, 6 * sumdiv(n, d, (d%5>0) * d))}; /* Michael Somos, Feb 04 2006 */

PARI

{a(n) = my(G); if( n<0, 0, G = [ 2, 1, 0, 0; 1, 2, 1, 0; 0, 1, 4, 5; 0, 0, 5, 10]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n, 1)), n))}; /* Michael Somos, Jun 12 2014 */

Sage

ModularForms( Gamma0(5), 2, prec=70).0;  # Michael Somos, Jun 12 2014

Formula

a(n) = 6*b(n) where b(n) is multiplicative with a(0) = 1, b(5^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise. - Michael Somos, Feb 04 2006

G.f. 1 + 6 * (Sum_{k>0} k * x^k / (1 - x^k) - 5*k * x^(5*k) / (1 - x^(5*k))). - Michael Somos, Feb 04 2006