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

0,2

References

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, 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, Trans. Amer. Math. Soc., Vol. 368, No. 11 (2016), pp. 7883-7910; ResearchGate preprint, 2015.

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

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

Sum_{k=0..n} a(k) ~ (2*Pi^2/5) * n^2. - Amiram Eldar, Nov 24 2025

Example

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 + ...

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 *)

Prog

(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 */
(SageMath) ModularForms( Gamma0(5), 2, prec=70).0;  # Michael Somos, Jun 12 2014
(Magma) Basis( ModularForms( Gamma0(5), 2), 70) [1]; /* Michael Somos, Jun 12 2014 */

Crossrefs

Sequence in context: A236864 A372566 A101527 * A283118 A274536 A051395

Adjacent sequences: A028884 A028885 A028886 * A028888 A028889 A028890

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved